Download Chapter 3: Topological Spaces

Chapter III
Topological Spaces
1. Introduction
In Chapter I we looked at properties of sets, and in Chapter II we added some additional structure
to a set  a distance function .  to create a pseudometric space. We then looked at some of the
most basic definitions and properties of pseudometric spaces. There is much more, and some of
the most useful and interesting properties of pseudometric spaces will be discussed in Chapter IV.
But in Chapter III we look at an important generalization.
Early in Chapter II we observed that the idea of continuity (in calculus) depends on talking about
“nearness,” so we used a distance function . to make the idea of “nearness” precise. The result
was that we could carry over the definition of continuity from calculus to pseudometric spaces.
The distance function . also led us to the idea of an open set in a pseudometric space. From there
we developed properties of closed sets, closures, interiors, frontiers, dense sets, continuity, and
sequential convergence.
One important observation was that open (or closed) sets are all we need to work with many of
these concepts; that is, we can often do what we need using the open sets without knowing what
specific . that generated these open sets: the topology g. is what really matters. For example,
cl E is defined in Ð\ß .Ñ in terms of the closed setsß so cl E doesn't change if . is replaced with a
different but equivalent metric . w  one that generates the same open sets. Changing . to an
equivalent metric . w also doesn't affect interiors, continuity, or convergent sequences. In
summary: for many purposes . is logically unnecessary (although . might be a handy tool) after
. has done its job in creating the topology g. Þ
This suggests a way to generalize our work. For a particular set \ we can simply add a
topology  that is, a collection of “open” sets given without any mention of a pseudometric that
might have generated them. Of course when we do this, we want these “open sets” to “behave
the way open sets should behave.” This leads us to the definition of a topological space.
2. Topological Spaces
Definition 2.1 A topology g on a set \ is a collection of subsets of \ such that
i) gß \ − g
ii) if Sα − g for each α − Eß then +−E S+ − g
iii) if S" ß ÞÞÞß S8 − g ß then S" ∩ ÞÞÞ ∩ S8 − g Þ
A set S © \ is called open if S − g . The pair Ð\ß g Ñ is called a topological space.
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We emphasize that in a topological space there is no distance function . : therefore concepts like
“distance between two points” and “%-ball” make no sense in Ð\ß g ÑÞ There is no preconceived
idea about what “open” means; to say “S is open” means nothing more or less than “S − g .”
In a topological space Ð\ß g Ñ, we can go on to define closed sets and isolated points just as we
did in pseudometric spaces.
Definition 2.2 A set J © Ð\ß g Ñ is called closed if X  J is open, that is, if \  J − g .
Definition 2.3 A point + − Ð\ß g Ñ is called isolated if Ö+} is open, that is, if Ö+× − g Þ
The proof of the following theorem is the same as it was for pseudometric spaces; we just take
complements and apply properties of open sets.
Theorem 2.4 In any topological space Ð\ß g Ñ
i) g and \ are closed
ii) if Jα is closed for each + − Eß then α−E Jα is closed
iii) if J" ß ÞÞÞß J8 are closed, then 83œ3 J3 is closed.
More informally, ii) and iii) state that intersections and finite unions of closed sets are closed.
Proof Read the proof for Theorem II.4.2. ñ
For a particular topological space Ð\ß g Ñ, it is sometimes possible to find a pseudometric . on \
for which g. œ g  that is, a . which generates exactly the same open sets as those already
given in g . But this cannot always be done.
Definition 2.5 A topological space Ð\ß g Ñ is called pseudometrizable if there exists a
pseudometric . on \ such that g. œ g Þ If . is a metric, then Ð\ß g Ñ is called metrizable.
Examples 2.6
1) Suppose \ is a set and g œ Ögß \×Þ g is called the trivial topology on \ and it is the
smallest possible topology on \ . Ð\ß g Ñ is called a trivial topological space. The only open (or
closed) sets are g and \Þ
If we put the trivial pseudometric . on \ , then g. œ g Þ So a trivial topological space
turns out to be pseudometrizable.
At the opposite extreme, suppose g œ c Ð\Ñ. Then g is called the discrete topology on
\ and it is the largest possible topology on \Þ Ð\ß g Ñ is called a discrete topological space.
Every subset is open (and also closed). Every point of \ is isolated.
If we put the discrete unit metric . (or any equivalent metric) on \ , then g. œ g Þ So a
discrete topological space is metrizable.
2) Suppose \ œ Ö!ß "× and let g œ Ögß Ö"×ß \×. Ð\ß g Ñ is a topological space called
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Sierpinski space. In this case it is not possible to find a pseudometric . on \ for which g. œ g ,
so Sierpinski space is not pseudometrizable. To see this, consider any pseudometric . on \ .
If .Ð!ß "Ñ œ !, then . is the trivial pseudometric on \ and Ögß \× œ g. Á g .
If .Ð!ß "Ñ œ $  !, then the open ball F$ Ð!Ñ œ Ö!× − g. ß so g. Á g Þ
(In this case g. is actually the discrete topology: . is just a rescaling of the
discrete unit metric.)
Another possible topology on \ œ Ö!ß "} is g w œ Ögß Ö!×ß \×, although Ð\ß g Ñ and Ð\ß g w Ñ
seem very much alike: both are two-point spaces, each with containing exactly one isolated
point. One space can be obtained from the other simply renaming “!” and “"” as “"” and “!”
respectively. Such “topologically identical” spaces are called “homeomorphic.” We will give a
precise definition of what this means later in this chapter.
3) For a set \ , let g œ ÖS © \ À S œ g or \  S is finite×Þ g is a topology on \ À
i) Clearly, g − g and \ − g .
ii) Suppose Sα − g for each α − EÞ If α−E Sα œ gß then α−E Sα − g .
Otherwise there is at least one Sα! Á g. Then \  Sα! is finite, so \  α−E Sα
œ α−E Ð\  Sα Ñ © \  Sα! . Therefore \  α−E Sα is finite, so α−E Sα − g .
iii) If S" ß ÞÞÞß S8 − g and some S4 œ g, then 83œ" S3 œ g so 83œ" S3 − g .
Otherwise each S3 is nonempty, so each \  S3 is finiteÞ Then Ð\  S" Ñ ∪ ÞÞÞ ∪ Ð\  S8 Ñ
œ \  83œ" S3 is finite, so 83œ" S3 − g .
In Ð\ß g Ñ, a set J is closed iff J œ g or J is finite. Because the open sets are g and the
complements of finite sets, g is called the cofinite topology on \ .
If \ is a finite set, then the cofinite topology is the same as the discrete topology on \ . (Why? )
In \ is infinite, then no point in Ð\ß g Ñ is isolated.
Suppose \ is an infinite set with the cofinite topology g Þ If Y and Z are nonempty open sets,
then \  Y and \  Z must be finite so Ð\  Y Ñ ∪ Ð\  Z Ñ œ \  ÐY ∩ Z Ñ is finite. Since
\ is infinite, this means that Y ∩ Z Á g Ðin fact, Y ∩ Z must be infiniteÑ. Therefore every pair of
nonempty open sets in Ð\ß g Ñ has nonempty intersection!
The preceding paragraph lets us see that an infinite cofinite space Ð\ß g Ñ is not
pseudometrizable:
i) if . is the trivial pseudometric on \ , then certainly g. Á g , and
ii) if . is not the trivial pseudometric on \ , then there exist points
+ß , − \ for which .Ð+ß ,Ñ œ $  !. In that case, F $ Ð+Ñ and F $ Ð,Ñ
#
#
would be disjoint nonempty open sets in g. ß so g. Á g Þ
4) On ‘, let g œ ÖÐ+ß ∞Ñ: + − ‘× ∪ Ög, ‘×. It is easy to verify that g is a topology on
‘, called the right-ray topology. Is Ð‘ß g Ñ metrizable or pseudometrizable?
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If \ œ g, then g œ Ög× is the only possible topology on \ , and g œ Ögß Ö+×× is the only
possible topology on a singleton set \ œ Ö+×. But for l\l  ", there are many possible
topologies on \ . For example, there are four possible topologies on the set \ œ Ö+ß ,×. These
are the trivial topology, the discrete topology, Ögß Ö+×ß \×ß and Ögß Ö,×ß \× although the last
two, as we mentioned earlier, can be considered as “topologically identical.”
If g is a topology on \ , then g is a collection of subsets of \ß so g © c Ð\Ñ. This means that
l\l
g − c Ðc Ð\ÑÑ, so |c Ðc Ð\ÑÑ| œ #Ð# Ñ is an upper bound for the number of possible topologies
(
on \ . For example, there are no more than ## ¸ $Þ% ‚ "!$) topologies on a set \ with 7
elements. But this upper bound is actually very crude, as the following table (given without
proof) indicates:
8 œ l\l
Actual number of
topologies on \
0
1
2
3
4
5
6
7
1
1
4
29
355
6942
209527
9535241 (many less than "!$) )
Counting topologies on finite sets is really a question about combinatorics and we will not pursue
that topic.
Each concept we defined for pseudometric spaces can be carried over directly to topological
spaces if the concept was defined in topological terms  that isß in terms of open (or closed) sets.
This applies, for example, to the definitions of interior, closure, and frontier in pseudometric
spaces, so these definitions can also be carried over verbatim to a topological space Ð\ß g ÑÞ
Definition 2.7 Suppose E © Ð\ß g Ñ. We define
int\ E œ the interior of E in \ œ ÖS À S is open and S © E×}
cl\ E œ the closure of E in \ œ ÖJ À J is closed and J ª E×
Fr\ E œ the frontier (or boundary) of E in \ œ cl\ E ∩ cl\ Ð\  EÑ
As before, we will drop the subscript “\ ” when the context makes it clear.
The properties for the operators cl, int, and Fr (except those that mention a pseudometric . or
an %-ball) remain true. The proofs in the preceding chapter were deliberately phrased in
topological terms so they would carry over to the more general setting of topological spaces.
Theorem 2.8 Suppose E © Ð\ß .Ñ. Then
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1) a) int E is the largest open subset of E (that is, if S is open and O © E, then
S © int E).
b) E is open iff E œ int E (since int E © E, the equality is equivalent to
E © int EÑ.
c) B − int E iff there is an open set S such that B − S © E
2) a) cl E is the smallest closed set containing E (that is, if J is closed and J ª E,
then J ª cl E ).
b) E is closed iff E œ cl E (since E © cl Eß the equality is equivalent to
cl E © E)Þ
c) B − cl E iff for every open set S containing B, S ∩ E Á g
3) a) Fr E is closed and Fr E œ Fr (X  E).
b) B − Fr E iff for every open set S containing B, S ∩ E Á g and
S ∩ Ð\  EÑ Á g
cÑ E is clopen iff Fr E œ g.
See the proof of Theorem II.4.5
At this point, we add a few additional facts about these operators. Some of the proofs are left as
exercises.
Theorem 2.9 Suppose E, F are subsets of a topological space Ð\ß g ÑÞ Then
1) cl ÐE ∪ F Ñ œ cl ÐEÑ ∪ cl ÐF Ñ
2) cl E œ E ∪ Fr E
3) int E œ E  Fr E œ \  clÐ\  EÑ
4) Fr E œ cl E  int E
5) \ œ int E ∪ Fr E ∪ int ÐX  EÑ, and these 3 sets are disjoint.
Proof 1) E © E ∪ F so, from the definition of closure, we have cl E © cl ÐE ∪ FÑ.
Similarly, cl F © cl ÐE ∪ FÑ Therefore cl E ∪ cl F © cl ÐE ∪ FÑÞ
On the other hand, cl E ∪ cl F is the union of two closed sets, so cl E ∪ cl F is closed
and cl E ∪ cl F ª E ∪ F , so cl E ∪ cl F ª clÐE ∪ FÑ. (As an exercise, try proving 1) instead
using the characterization of closures given above in Theorem 2.8.2c.)
Is 1) true if “ ∪ ” is replaced by “ ∩ ” ?
2) Suppose B − cl E but B Â EÞ If S is any open set containing B, then S ∩ E Á g
(because B − cl E) and S ∩ Ð\  EÑ Á g Ðbecause the intersection contains B). Therefore
B − Fr E, so B − A ∪ Fr AÞ
Conversely, suppose B − E ∪ Fr E. If B − E, then B − cl EÞ And if B Â E, then
B − Fr E œ cl E ∩ cl Ð\  EÑ, so B − cl EÞ Therefore cl A œ A ∪ Fr AÞ
The proofs of 3)  5) are left as exercises. ñ
Theorem 2.9 shows us that complements, closures, interiors and frontiers are interrelated and
therefore some of these operators are redundant. That is, if we wanted to very “economical,” we
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could discard some of them. For example, we could avoid using “Fr” and “int” and just use “cl”
and complement because Fr E œ cl E ∩ cl Ð\  EÑ and int E œ E  Fr E
œ E  Ðcl E ∩ cl Ð\  EÑÑ. Of course, the most economical way of doing things is not
necessarily the most convenient. (Could we get by only using complements and “Fr”  that is,
can we define “int” and “”cl” in terms of “Fr” and complements? Or could
we use just “int” and complements? )
Here is a famous related problem from the early days of topology: for E © Ð\ß g Ñ, is there an
upper bound for the number of different subsets of \ which might created from E using only
complements and closures, repeated in any order? (As we just observed, using the interior and
frontier operators would not help to create any additional sets.) For example, one might start
with E and then consider such sets as cl E, \  clEß clÐ\  clÐ\  EÑÑß and so on. An old
theorem of Kuratowski (1922) says that for any set E in any space Ð\ß g Ñ, the upper bound is 14.
Moreover, this upper bound is “sharp”  there is a set E © ‘ from which 14 sets can actually be
obtained! Can you find such a set?
Definition 2.10 Suppose H © Ð\ß g Ñ. H is called dense in \ if cl H œ \Þ The space Ð\ß g Ñ
is called separable if there exists a countable dense set H in \ .
Example 2.11 Let g be the cofinite topology on ‘. Let „ œ Ö#ß %ß 'ß ÞÞÞ×Þ
int „ œ g, because any nonempty open set S has finite complement and therefore is
not a subset of „. In fact, int F œ g whenever ‘  F is infinite.
cl „ œ ‘ because the only closed set containing „ is ‘. Therefore Ð‘ß g Ñ is separable.
In fact, any infinite set F is dense.
Fr „ œ cl Ð„Ñ ∩ cl Б  „Ñ œ ‘ ∩ ‘ œ ‘
Ð‘ß g Ñ is not pseudometrizable ( why? )
Example 2.12 Let g be the right-ray topology on ‘. In (‘ß g Ñ,
int ™ œ g
cl ™ œ ‘, so Ð‘ß g Ñ is separable
Fr ™ œ ‘
Any two nonempty open sets intersect, so Ð‘ß g Ñ is not metrizable. Is it
pseudometrizable?
3. Subspaces
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If E is a subset of a space Ð\ß g Ñ, then there is a natural way to make E into a topological space.
We use the open sets in \ to define open sets in EÞ
Definition 3.1 Suppose E © \ , where Ð\ß g Ñ is a topological space. The subspace topology on
E is defined as gE œ ÖE ∩ S À S − g }. ÐEß gE Ñ is called a subspace of Ð\ß g ÑÞ
(Check that gE is actually a topology on E. To say that E is a subspace of Ð\ß g Ñ implies
that E is given the topology gE Þ To emphasize that E is a subspace, we sometimes write
E © Ð\ß g Ñ rather than just E © \ ÞÑ
If E © \ , we sometimes refer to S ∩ E as the “restriction of S to E” or the “trace of S on E”
Example 3.2 Consider  © ‘, where ‘ has its usual topology. For each 8 − , the interval
Ð8  "ß 8  "Ñ is open in ‘. Therefore Ð8  "ß 8  "Ñ ∩  œ Ö8× is open in the subspace , so
every point 8 is isolated in the subspace. The subspace topology is the discrete topology. Notice:
the subspace topology on  is just what we would get if we used the usual metric on  to
generate open sets in . Similarly, it is easy to check that in ‘# the subspace topology on the
B-axis is the same as the usual metric topology on ‘.
Suppose E © Ð\ß .Ñ. Then we can think of two ways to make E into a topological space.
i) . gives a topology g. on \ . Take the open sets in g. and intersect them
with E. This gives us the subspace topology on E, which we call Ðg. ÑE and
ÐEß Ðg. ÑE Ñ is a subspace of Ð\ß g ÑÞ
2) Or, we could treat E as a pseudometric space: just use . to measure distances
in E. To be very precise, . À \ ‚ \ Ä ‘, so we make a “new” pseudometric
. w defined by . w œ .lE ‚ E. Then ÐEß .w Ñ is a pseudometric space and we can
use . w to generate open sets in E: the topology g.w Þ (Usually, we would be less
compulsive about the notation and just use . also to refer to the pseudometric
n EÞ But for just a moment it will help to distinguish . and . w œ .lE ‚ E
with different names.)
It turns out (fortunately) that 1) and 2) produce the same open sets in E À the open sets in ÐEß . w Ñ
are just the open sets from Ð\ß .Ñ restricted to EÞ That's just what Theorem 3.3 says (in “fancier”
notation).
Theorem 3.3 Suppose E © Ð\ß .Ñ, where . is a pseudometric: then Ðg. ÑE œ g. w .
Proof Suppose Y − Ðg. ÑE , so Y œ S ∩ E where S − g. . Let E − Y . There is an %  ! such
w
that F%. Ð+Ñ © SÞ Since . w œ .lE ‚ E, we get that F%. Ð+Ñ œ F%. Ð+Ñ ∩ E © S ∩ E œ Y ,
so Y − g. w .
w
Conversely, suppose Y − g. w Þ For each + − Y there is an %+  ! such that F%.+ Ð+Ñ © Y ß
w
w
and Y œ +−Y F%.+ Ð+ÑÞ Let S œ +−Y F%.+ Ð+Ñß an open set in g. Þ Since F%.+ Ð+Ñ œ F%.+ Ð+Ñ ∩ E,
we get S ∩ E œ Ð+−S F%.+ Ð+ÑÑ ∩ E œ Ð+−S F%.+ Ð+Ñ ∩ EÑ
w
œ +−Y F%.+ Ð+Ñ œ Y Þ Therefore Y − Ðg. ÑE Þ ñ
Exercise Verify that in any space Ð\ß g Ñ
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i) If Y is open in Ð\ß g Ñ and E is an open set in the subspace Y ß then E is open in \Þ
(“An open subset of an open set is open.” )
ii) If J is closed in Ð\ß g Ñ and E is a closed set in the subspace J ß then J is closed
in \ . (“A closed subset of a closed set is closed.” )
4. Neighborhoods
Definition 4.1 If R © Ð\ß g Ñ and B − int R , then we say that N is a neighborhood of B Þ
The collection aB œ ÖR © \ À R is a neighborhood of B× is called the neighborhood system
at BÞ
Note that
1) aB Á g, because every point B has at least one neighborhood  for example,
R œ \Þ
2) If R" and R# − aB , then B − int R" ∩ int R# œ (why?) int ÐR" ∩ R# Ñ.
Therefore R" ∩ R# − aB .
3) If R − aB and R © R w , then B − int R © int R w , so R w − aB (that is, if
R w contains a neighborhood of B, then R w is also a neighborhood of B.)
Just as in pseudometric spaces, it is clear that a set S in Ð\ß g Ñ is open iff O is a neighborhood of
each of its points.
In a pseudometric space, we use the %-balls centered at B to measure “nearness” to B. For
example, saying that “every %-ball in ‘ centered at B contains an irrational number” tells us that
“there are irrational numbers arbitrarily near to B.” Of course, we could convey the same
information in terms of neighborhoods by saying “every neighborhood of B in ‘ contains an
irrational number.” Or instead we could say it in terms of “open sets”: “every open set in ‘
containing B contains an irrational number.” These are all equivalent ways to say “there are
irrational numbers arbitrarily near to B.” That we can say it all these different ways isn't
surprising since, in Ð\ß .Ñß open sets and neighborhoods were defined in terms of %-balls.
In a topological space Ð\ß g Ñ we don't have %-balls, but we still have open sets and
neighborhoods. We now think of the neighborhoods in aB (or, if we prefer, the collection of
open sets containing B) as the tool we use to talk about “nearness to B.”
For example, suppose \ has the trivial topology g . For any B − \ß the only neighborhood of B
is \ : therefore every C in \ is in every neighborhood of B À the neighborhoods of B are unable
to “separate” B and C and that's analogous to having .ÐBß CÑ œ ! (if we had a pseudometric). In
that sense, all points in Ð\ß g Ñ are “very close together”: so close together, in fact, that they are
“indistinguishable.” The neighborhoods of B tell us this.
At the opposite extreme, suppose \ has the discrete topology g and that B − \Þ If B − [ then
(since [ is open), [ is a neighborhood of B. R œ ÖB× is the smallest neighborhood of B. So
every point B has a neighborhood R that excludes all other points C: for every C Á B, we could
say “C is not within the neighborhood R of B.” This is analogous to saying “C is not within % of
B” (if we had a pseudometric). Because no point C is “within R of Bß ” we call B isolated. The
neighborhoods of B tell us this.
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(Of course, if we prefer, we could use “Y is an open set containing B” instead of
“R is a neighborhood of B” to talk about nearness to B.)
The complete neighborhood system aB of a point B often contains more neighborhoods than we
actually need to talk about nearness to B. For example, using just the (open) neighborhoods
F " ÐBÑ in a pseudometric space Ð\ß .Ñ is enough to let us talk about continuous functions at B.
8
We use the idea of a neighborhood base at B to choose a smaller collection of neighborhoods of B
that is i) good enough for all our purposes, and ii) from which all the other neighborhoods of B
can be obtained if we want them.
Definition 4.2 A collection UB © aB is called a neighborhood base at B if for every
neighborhood R of B, there is a neighborhood F − UB such that B − F © R . We refer to the
sets in UB as basic neighborhoods of B.
According to the definition, each set in UB must be a neighborhood of B, but the collection UB
may be much simpler than the whole neighborhood system RB Þ The crucial thing is that every
neighborhood R of B must contain “basic neighborhood” F of B.
Example 4.3 At a point B − Ð\ß .Ñ, let
i)
UB œ the collection of all balls F% ÐBÑ
ii) UB œ the collection of all balls F% ÐBÑ , where % is a positive rational
iii) UB œ the collection of all balls F " ÐBÑ for n − 
8
iv) UB œ aB
(The neighborhood system is always a base for itself, but it
is not an “efficient” choice; the goal is to get a base UB
that's much simpler than RB .)
Which UB to use is our choice: each of i)-iv) gives a neighborhood base at B. But ii) and iii)
might be more convenient: each of those gives a countable family UB as a neighborhood base. (If
\ œ ‘, for example, with the usual metric . , then the collections UB in i) and iv) are uncountable
collections.) Of the four, iii) is probably the simplest choice for UB .
Suppose we want to check whether some property that involving neighborhoods of B is true.
Often all we need to do is to check whether the property holds for the simpler neighborhoods in
UB . For example, in Ð\ß .Ñ À S is open iff O contains a neighborhood R of each B − SÞ But
that is true iff S contains a set F " ÐBÑ from UB .
8
Similarly, B − cl E iff R ∩ A Á g for every R − aB iff F ∩ E Á g for every F − UB . For
example, suppose we want to check, in ‘, that " − cl . It is sufficient just to check that
F " Ð"Ñ ∩  Á g for each 8 − , because this implies that R ∩  Á g for every R − a" Þ
8
Therefore an “efficient” choice of the “simplest possible” neighborhood base UB at each point B
is desirable.
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Definition 4.4 We say that a space Ð\ß g Ñ satisfies the first axiom of countability (or, more
simply, that Ð\ß g Ñ is first countable) if at each point B − \ , it is possible to choose a countable
neighborhood base UB Þ
Example 4.5
1) The preceding Example 4.3 shows that every pseudometric space is first countable.
2) If g is the discrete topology on \ß then Ð\ß g Ñ is first countable. In fact, at each
point B, we can choose a neighborhood base consisting of a single set: UB œ Ö ÖB× ×Þ
3) Let g be the cofinite topology on an uncountable set \ . For any B − \ , there cannot
be a countable neighborhood base UB at B.
To see this, suppose that we had a countable neighborhood base at B À
UB œ ÖF" ß ÞÞÞß F8 ß ÞÞÞ×Þ For any C Á Bß ÖC× is closedß so \  ÖC× is a
neighborhood of B. Therefore for some 5 , B − int F5 © F5 © \  ÖC×, so
∞
∞
C Â ∞
8œ" int F8 . But B −
8œ" int F8 so
8œ" int F8 œ ÖB×.
Taking complements, we get \  ÖB× œ \  ∞
8œ" int F8
œ ∞
Ð\

int
F
Ñ
.
Since
\

int
F
is
finite
(why?), this would mean that
8
8
8œ"
\  ÖB× is countable  which is impossible.
Since any pseudometric space is first countable, the example gives us another way to see that this
space Ð\ß g Ñ is not pseudometrizable.
For a given topology g on \ , all the neighborhood systems aB are completely determined by the
topology g , but UB is not. As the preceding examples illustrate, there are usually many possible
choices for UB . (Can you describe all the spaces Ð\ß g Ñ for which UB is uniquely determined at
each point B ?)
On the other hand, if we were given UB at each point B − \ we could
1) “reconstruct” the whole neighborhood system aB :
aB œ ÖR © \ À bF − UB such that B − F © R ×, and then we could
2) “reconstruct” the whole topology g :
g œ ÖS À S is a neighborhood of each of its points×
œ ÖS À aB − S bF − UB B − F © S×, that is,
S is open iff S contains a basic neighborhood of each of its points.
112
5. Describing Topologies
How can a topological space be described? If \ œ Ö!ß "×, it is simple to give a topology by just
writing g œ Ögß Ö!×ß \×Þ However, describing all the sets in g explicitly is often not the easiest
way to go.
In this section we look at three important, alternate ways to define a topology on a set. All of
these will be used throughout the remainder of the course. A fourth method  by using a
“closure operator”  is not used much nowadays. It is included just as an historical curiosity.
A. Basic Neighborhoods
Suppose that at each point B − Ð\ß g Ñ we have picked a neighborhood base UB . As mentioned
above, the collections UB contain implicitly all the information about the topology: a set S is in g
iff O contains a basic neighborhood of each of its points. This suggests that if we start with a set
\ , then we could define a topology on \ if we begin by saying what the UB 's should be. Of
course, we can't just put “random” sets in UB À the sets in each UB must “act the way basic
neighborhoods are supposed to act.” And how is that? The next theorem describes the crucial
behavior of a collection of basic neighborhoods at B.
Theorem 5.1 Suppose Ð\ß g Ñ is a topological space and that UB is a neighborhood base at B for
each B − \ . Then
1) UB Á g and F − UB Ê B − F © \
2) if F" and F# − UB , then b F$ − UB such that B − F$ © F" ∩ F#
3) if F − UB , then b M such that B − M © F and,
a C − I, b BC − UC such that C − BC © I
4) O − g Í aB − O b B − UB such that B − B © O.
Proof 1) Since \ is a neighborhood of B, there is a F − UB such that B − F © \ . Therefore
UB Á g. If F − UB © aB , then B is a neighborhood of B, so B − int F © F .
2) The intersection of the two neighborhoods F" and F# of B is again a
neighborhood of B. Therefore, by the definition of a neighborhood base, there is a set F$ − UB
such that B − F$ © F" ∩ F# .
3) Let M œ int F . Then B − I © B and because I is open, I is a neighborhood of each its
points C. Since UC is a neighborhood base at C, there is a set FC − UC such that C − FC © M .
4) Ê : If O is open, then O is a neighborhood of each of its points B. Therefore for each
B − S there must be a set F − UB such that B − F © SÞ
É : The condition implies that O contains a neighborhood of each of its points.
Therefore S is a neighborhood of each of its points, so S is open. ñ
113
The features of a neighborhood base in Theorem 5.1 are the crucial ones about the behavior of a
neighborhood base at B. The next theorem tells us that, with this information, we can put a
topology on a set by giving a collection of sets to become the basic neighborhoods at each point
B.
Theorem 5.2 (The Neighborhood Base Theorem) Let \ be a set. Suppose that for each
B − \ we assign a collection UB of subsets of \ in such a way that conditions 1) - 3) of Theorem
5.1 are true. Define g œ {O © \ : aB − O b F − UB such that B − F © O}Þ
Then g is a topology on \ and UB is now a neighborhood base at B in Ð\ß g ÑÞ
Note: In Theorem 5.2, we do not ask that the UB 's satisfy condition 4) of Theorem 5.1  because
the set \ has no topology and condition 4) would be meaningless. Here, Condition 4) is our
motivation for how to define g using the FB 's.
Proof We need to prove three things: a) that g is a topology, that b) in Ð\ß g Ñ, each F − UB
is now a neighborhood of B, and that c) the collection UB is now a neighborhood base at B.
a) Clearly, g − g . If B − \ then, by condition 1), we can choose a F − UB and
B − F © \Þ Therefore \ − g .
Suppose Oα − g for all α − E. If B − {Oα : α − E}, then B − Oα! for some
α! − E. By definition of g , there is a set F − UB such that B − B © Oα! © {Oα : α − E}, so
{Oα : α − E} − g .
To finish a), it is sufficient to show that if S" and S# − g , then S" ∩ S# − g .
Suppose B − S" ∩ S# . By the definition of g , there are sets F" and F# − UB such that
B − F" © S" and B − F# © S# , so B − F" ∩ F# © S" ∩ S# . By condition 2), there is a set
F$ − UB such that B − F$ © F" ∩ F# © O" ∩ O# . Therefore O" ∩ O# − g .
Therefore g is a topology on \  so now we have a topological space Ð\ß g Ñ  and we must
show that in this space UB is now a neighborhood base at B. Doing that involves the awkwardlooking condition 3)  which we have not yet used.
b) If F − UB , then B − F (by condition 1) and (by condition 3), there is a set M © X such
that B − M © F and a C − M , b FC − UC such that C − FC © M . The underlined phrase states that
M satisfies the condition for M − g , so M is open. Since M is open and B − M © F , F is a
neighborhood of B, that is, UB © aB .
c) To complete the proof, we have to check that UB is a neighborhood base at B. If
R is a neighborhood of B, then B − int R . Since int R is open, int R must satisfy the criterion
for membership in g , so there is a set F − UB such that B − F © int R © R . Therefore UB
forms a neighborhood base at B. ñ
Example 5.3 For each B − ‘, let UB œ ÖÒBß ,Ñ À ,  B×. We can easily check the conditions
1) - 3) from Theorem 5.2:
114
1) For each B − ‘, certainly UB Á g and B − ÒBß ,Ñ for each set ÒBß ,Ñ − UB
2) If F" œ ÒBß ," Ñ and F# œ ÒBß ,# Ñ are in UB , then (in this example) we can
choose F$ œ F" ∩ F# œ ÒBß ,$ Ñ − UB , where ,$ œ min Ö," ß ,# ×Þ
3) If F œ ÒBß ,Ñ − UB , then (in this example) we can let M œ FÞ
If C − M œ ÒBß ,Ñß pick - so C  -  ,. Then FC œ ÒCß -Ñ − UC and
C − ÒCß -Ñ © MÞ
According to Theorem 5.2, g œ ÖS © ‘ À aB − S b ÒBß ,Ñ − UB such that B − ÒBß ,Ñ © S× is a
topology on ‘ and UB is a neighborhood base at B in Ð‘ß g Ñ. The space Ð‘ß g Ñ is called the
Sorgenfrey line.
Notice that in this example each set ÒBß ,Ñ − g : that is, the sets in UB turn out to be open (not
merely neighborhoods of BÞ ÐThis does not always happen.Ñ
It is easy to check that sets of form Ð  ∞ß BÑ and Ò,ß ∞Ñ are open, so Ð  ∞ß BÑ ∪ Ò,ß ∞Ñ
œ ‘  ÒBß ,Ñ is open. Therefore ÒBß ,Ñ is also closed. So, in the Sorgenfrey line, there is a
neighborhood base UB consisting of clopen sets at each point BÞ
"
We can write Ð+ß -Ñ œ ∞
8œ" Ò+  8 ß -Ñ , so Ð+ß -Ñ is open in the Sorgenfrey line. Because every
usual open set in ‘ is a union of sets of the form Ð+ß -Ñ, we conclude that every usual open set in
‘ is also open in the Sorgenfrey line. The usual topology on ‘ is strictly smaller that the
Sorgenfrey topology: g. § g .
Á
 is dense in the Sorgenfrey line: if B − ‘ , then every basic neighborhood ÒBß ,Ñ of B intersects
, so B − cl Þ Therefore the Sorgenfrey line is separable. It is also clear that the Sorgenfrey
line is first countable: at each point B the collection ÖÒBß B  8" Ñ À 8 − × is a countable
neighborhood base.
Example 5.4 Similarly, we can define the Sorgenfrey plane by putting a new topology on ‘# .
At each point ÐBß CÑ − ‘# , let UÐBßCÑ œ ÖÒBß ,Ñ ‚ ÒCß -Ñ À ,  Bß -  C ×. The families UÐBßCÑ
satisfy the conditions in the Neighborhood Base Theorem, so they give a topology g for which
UÐBßCÑ is a neighborhood base at ÐBß CÑÞ A set S © ‘# is open iff: for each ÐBß CÑ − S, there are
,  B and -  C such that ÐBß CÑ − ÒBß ,Ñ ‚ ÒCß -Ñ © SÞ ÐMake a sketch!Ñ You should check that
the sets ÒBß ,Ñ ‚ ÒCß -Ñ − UÐBßCÑ are actually clopen in the Sorgenfrey plane. It is also easy to
check the usual topology g. on the plane is strictly smaller that the Sorgenfrey topology. It is
clear that # is dense, so Б# ß g Ñ is separable. Is the Sorgenfrey plane first countable?
Example 5.5 At each point : − ‘# , let G% Ð:Ñ œ ÖB − ‘# À .ÐBß :Ñ Ÿ %× and define
U: œ ÖG% Ð:Ñ À : − ‘# ×. It is easy to check that the conditions 1) - 3) of Theorem 5.2 are
satisfied. ÐFor G% Ð:Ñ in condition 3), let I œ F% Ð:ÑÞÑ The topology generated by the U: 's is just
the usual topologyà the sets in U: are basic neighborhoods of : in the usual topology  as they
should be  but the sets in U: did not turn out to be open sets.
115
Example 5.6 Let > œ ÖÐBß CÑ − ‘# À C !× œ the “closed upper half-plane.”
For a point : œ ÐBß CÑ − > with C  !ß let U: œ ÖF% Ð:Ñ À %  l C l×
For a point : œ ÐBß !Ñ − >ß let
U: œ ÖÖ:× ∪ E À E is a usual open disc in the upper half-plane, tangent to the B-axis at :×Þ
It is easy to check that the collections U: satisfy the conditions 1) - 3) of Theorem 5.2 and
therefore give a topology on >. In this topology, the sets in U: turn out to be open neighborhoods
of :.
The space >, with this topology, is called the “Moore plane.” Notice that > is separable and 1st
countable. The subspace topology on the B-axis is the discrete topology. (Verify these
statements! )
B. Base for the topology
Definition 5.7 A collection of open sets in Ð\ß g Ñ is called a base for the topology g if each
S − g is a union of sets from U Þ More precisely, U is a base if U © g and for each S − g ß
there exists a subfamily T © U such that S œ T. We also call U a base for the open sets and
we refer to the open sets in U as basic open sets.
If U is a base, then it is easy to see that: S − g iff aB − O bF − U such that B − F © O. This
means that if we were given U , we could use it to decide which sets are open and thus
“reconstruct” g .
Of course, we could choose U œ g : every topology g is a base for itself. But there usually are
many ways to choose a base, and the idea is that a simpler base U would be easier to work with.
For example, the set of all balls is a base for the topology g. in any pseudometric space Ð\ß .Ñà a
different base would be the set containing only the balls with positive rational radii.
(Can you describe those topological spaces for which g is the only base for g ?)
The following theorem tells us the crucial properties of a base U in Ð\ß g ÑÞ
Theorem 5.8 If Ð\ß g Ñ is a topological space with a base U for g , then
1) \ œ ÖF À F − U ×
2) if F" and F# − U and B − F" ∩ F# , then there is a set F$ − U
such that B − F$ © F" ∩ F# .
Proof 1) Certainly U © \Þ But by definition of base, the open set \ is a union of a subfamily
of U . Therefore U œ \Þ
2) If F" and F# − U , then F" ∩ F# is open. If B − F" ∩ F# , the definition of base implies
that there must be a set F$ − U such that B − F$ © F" ∩ F# . ñ
116
The next theorem tells us that if we are given a collection U of subsets of a set \ with properties
1) and 2), we can use it to define a topology.
Theorem 5.9 (The Base Theorem) Suppose \ is a set and that U is a collection of subsets of \
that satisfies conditions 1) and 2) in Theorem 5.8. Define g œ ÖO © \ À O is a union of sets
from U × œ ÖS © \ À aB − S bF − U such that B − F © S×.
Then g is a topology on \ and U is a base for g .
Proof First we show that g is a topology on \ . Since g is the union of the empty subfamily of
U , we get that g − g , and condition 1) simply states that \ − g .
If Oα − g (α − E), then ÖSα À α − E× is a union of sets Sα each of which is a union of sets
from U . So clearly ÖSα À α − E× is a union of sets from U . So ÖSα À α − E× − g .
Suppose O" and O# − g and that B − S" ∩ O# . For each such B we can use 2) to pick a set
FB − U such that B − FB © O" ∩ O# . Then O" ∩ O# is the union of all the BB 's chosen in this
way, so O" ∩ O# − g .
Now we know that we have a topology, g , on \ . By definition of g it is clear that U © g and
that each set in g is a union of sets from U Þ Therefore U is a base for g . ñ
Example 5.10 The collections
U œ ÖF: Ð%Ñ À : − ‘# , %  ! ×
U w œ ÖF: Ð 5" Ñ À : − # , 5 −  × and
ww
U œ ÖÐ+ß ,Ñ ‚ Ð-ß .Ñ À +ß ,ß -ß . − ‘, +  ,ß -  .×
each satisfy the conditions 1) - 2) in Theorem 5.9, so each collection is the base for a topology on
‘# Þ In fact, all three are bases for the same topology on ‘#  that is, the usual topology Ðcheck
this!Ñ Of the three, U w is the simplest choice  it is a countable base for the usual topology.
Example 5.11 Suppose Ð\ß g w Ñ and Ð] ß g ww Ñ are topological spaces. Let U œ the set of “open
boxes” in \ ‚ ] œ ÖY ‚ Z À Y − g w and Z − g ww ×Þ (Verify that U satisfies conditions 1) and
2) of The Base Theorem.) The product topology on the set \ ‚ ] is the topology for which U is
a base. We always assume that \ ‚ ] has the product topology unless something else is stated.
Therefore a set E © \ ‚ ] is open (in the product topology) iff À for all ÐBß CÑ − Eß there are
open sets Y © \ and Z © ] such that ÐBß CÑ − Y ‚ Z © EÞ ÐNote that E itself might not be a
“box.”Ñ
Let 1B À \ ‚ ] Ä \ be the “projection” defined by 1B ÐBß CÑ œ B. If Y is any open set in \ ,
then 1B" ÒY Ó œ Y ‚ ] − U . Therefore 1B" ÒY Ó is open. Similarly, for the other projection map
1C À \ ‚ ] Ä ] defined by 1C ÐBß CÑ œ C : if Z is open in ] , then 1C" ÒZ Ó œ \ ‚ Z is open in
\ ‚ ] Þ (As we see in Section 8, this means that the projection maps are continuous. It is not
hard to show that the projection maps 1B and 1C are open maps.: that is, the images aof open sets
in the product are open.)
117
If H" is dense in \ and H# is dense in ] , we claim that H" ‚ H is dense in \ ‚ ] . If
ÐBß CÑ − E, where E is open, then there are nonempty open sets Y © \ and Z © ] for which
ÐBß CÑ − Y ‚ Z © EÞ Since B − -6 D" , Y ∩ H" Á g; and similarly Z ∩ H# Á gÞ Therefore
ÐY ‚ Z Ñ ∩ ÐH" ‚ H# Ñ œ ÐY ∩ H" Ñ ‚ ÐZ ∩ H# Ñ Á g, so E ∩ ÐH" ‚ H# Ñ Á gÞ
Therefore
ÐBß CÑ − cl ÐH" ‚ H# ÑÞ So H" ‚ H# is dense in \ ‚ ] . In particular, this shows that the product
of two separable spaces is separable.
Example 5.12 The open intervals Ð+ß ,Ñ form a base for the usual topology in ‘, so each set
Ð+ß ,Ñ ‚ Ð-ß .Ñ is in the base U for the product topology on ‘ ‚ ‘. It is easy to see that every
“open box” Y ‚ Z in U can be written as a union of simple “open boxes” like Ð+ß ,Ñ ‚ Ð-ß .Ñ.
Therefore U w œ ÖÐ+ß ,Ñ ‚ Ð-ß .Ñ À +  ,ß -  .× also is a (simpler) base for the product topology
on ‘ ‚ ‘. From this, it is clear that the product topology on ‘ ‚ ‘ is the usual topology on the
plane ‘# (see Example 5.10).
In general, the open sets Y and Z in the base for the product topology on \ ‚ ] can be replaced
by sets “Y chosen from a base for \ ” and “Z chosen from a base for ] ,” as in this example.
So in the definition of the product topology, it is sufficient to say that basic open sets are of the
form Y ‚ Z ß where Y +8. V are basic open sets from \ and from ] Þ
Definition 5.13 A space Ð\ß g Ñ is said to satisfy the second axiom of countability (or, more
simply, to be a second countable space) if it is possible to find a countable base U for the
topology g .
For example, ‘ is second countable because, for example, U œ ÖÐ+ß ,Ñ À +ß , − × is a countable
base. Is ‘# is second countable (why or why not) ?
Example 5.14 The collection U œ ÖÒBß B  8" Ñ À B − ‘ß 8 − × is a base for the Sorgenfrey
topology on ‘. But the collection ÖÒBß B  8" Ñ À B − ß 8 − × is not a countable base for the
Sorgenfrey topology. Why not?
Since the sets in a base may be simpler than arbitrary open sets, they are often more convenient to
work with, and working with the basic open sets is often all that is necessary  not a surprise
since all the information about the open sets in contained in the base U . For example, you should
check that
1) If U is a base for g , then B − cl E iff each basic open set F containing B satisfies
F ∩ E Á gÞ
2) If 0 À Ð\ß .Ñ Ä Ð] ß . w Ñ and U is a base for the topology g. w on ] , then 0 is continuous
iff 0 " ÒFÓ is open for each F − U . This means that we needn't check the inverse images of all
open sets to verify that 0 is continuous.
118
C. Subbase for the topology
Definition 5.15 Suppose Ð\ß g Ñ is a topological space. A family Æ of open sets is called a
subbase for the topology g if the collection U containing all finite intersections of sets from Æ is
a base for g . (It is clear that if U is a base for g , then U is also a subbase for g .)
Examples i) The collection Æ œ ÖM À M œ Ð  ∞ß ,Ñ or M œ Ð+ß ∞Ñß +ß , − ‘× is a subbase for
a topology on ‘. All intervals of the form Ð+ß ,Ñ œ Ð  ∞ß ,Ñ ∩ Ð+ß ∞Ñ are in U, and U is a base
for the usual topology on ‘.
ii) The collection Æ of all sets Y ‚ ] and Z ‚ \ Ðfor Y open in \ and Z open in ] )
is a subbase for the product topology on \ ‚ ] : the collection U of finite intersections from Æ
includes all open boxes Y ‚ Z œ ÐY ‚ ] Ñ ∩ Ð\ ‚ Z Ñ.
We can define a topology on a set \ by giving a collection Æ of subsets as the subbase for a
topology. Surprisingly, any collection Æ can be used: no special conditions on Æ are required.
Theorem 5.16 (The Subbase Theorem) Suppose \ is a set and Æ is any collection of subsets
of \ . Let U be the collection of all finite intersections of sets from Æ. Then U is a base for a
topology g , and Æ is a subbase for g .
Proof
First we show that U satisfies conditions 1) and 2) of The Base Theorem.
1) \ − U since \ is the intersection of the empty subcollection of Æ (this follows the
convention that the intersection of an empty family of subsets of \ is \ itself. See Example
I.4.5.5 ). Since \ − U ß certainly \ œ {F : F − U }Þ
2) Suppose F" and F# − U , and B − F" ∩ F# . We know that F" œ S" ∩ ... ∩ S7
and F# œ S7" ∩ ... ∩ S75 for some S" ,..., S7 , ..., S75 − Æ, so
B − F$ œ F" ∩ F# œ W" ∩ ... ∩ W7 ∩ W7" ∩ ... ∩ W75 − U
Therefore g œ ÖS À S is a union of sets from U × is a topology, and U is a base for g . By
definition of U and g , we have Æ © U © g and each set in g is a union of finite intersections of
sets from Æ. Therefore Æ is a subbase for g . ñ
Example 5.17
1) Let „ œ Ö#ß %ß 'ß ÞÞÞ × ©  and Æ œ ÖÖ"×ß Ö#×ß „ ×Þ Æ is a subbase for a topology on
. A base for this topology is the collection U of all finite intersections of sets in Æ:
U œ Ög, ß Ö#×ß Ö"×ß „ ×.
and the (not very interesting) topology g generated the set of all possible unions of sets from U :
g œ Ög, , Ö"}, {2}, {1,2}, „, „ ∪ Ö"××
119
2) For each 8 − ß let W8 œ Ö8ß 8  "ß 8  #ß ÞÞÞ×Þ The collection Æ œ ÖW8 À 8 − × is
a subbase for a topology on . Here, W8 ∩ W7 œ W5 where 5 œ maxÖ7ß 8×, so the collection of
all finite intersections from Æ is just Æ itself. So U œ Æ is actually a base for a topology. The
topology is g œ Æ ∪ Ög×.
3) Let Æ œ Öj À j is a straight line in ‘# × in ‘# . If : − ‘# , then Ö:× is the intersection
of two sets from Æ, so Ö:× − U Þ Æ is a subbase for the discrete topology on ‘# .
4) Let Æ œ Öj À j is a vertical line in ‘# ×Þ Æ generates a topology on ‘# for which
U œ Ö‘ ß g } ∪ Æ is a base and g œ ÖS À S is a union of vertical lines×Þ
#
5) Suppose g w is any topology on \ for which Æ © g w Þ g w must contain the finite
intersections of sets in Æ, and therefore must contain all possible unions of those intersections.
Therefore g w contains the topology g for which Æ is a subbase. To put it another way, the
topology for which Æ is a subbase is in the smallest topology on \ containing the collection Æ.
In fact, g œ Ög w À g w ª Æ and g w is a topology on \×.
Caution We said earlier that for some purposes it is sufficient to work with basic open sets,
rather than arbitrary open sets  for example to check whether B − cl E, or to check whether a
function 0 between metric spaces is continuous. However, it is not always sufficient to work
with subbasic open sets. Some caution is necessary.
For example, Æ œ ÖM À M œ Ð  ∞ß ,Ñ or M œ Ð+ß ∞Ñß +ß , − ‘× is a subbase for the
usual topology on ‘. We have M ∩ ™ Á g for every subbasic open set M containing "# , but
"
# Â cl ™Þ
D. The closure operator
Usually we describe a topology g by giving a subbase Æ, a base U , or by giving collections UB to
be the basic neighborhoods at each point B. In the early history of general topology, one other
method was sometimes used. We will never use it, but we include it here as a curiosity.
Let clg be the closure operator in Ð\ß g Ñ (normally, we would just write “cl” for the closure
operator; here we write “clg ” to emphasize that this closure operator comes from the topology
g on \). It gives us all the information about g . That is, using clg ß we can decide whether any
set E is closed (by asking “is clg E œ E ? ”) and therefore can decide whether any set F is open
(by asking “is \  F closed ?”). It should be not be a surprise, then, that we can define a
topology on a set \ if we are start with an “operator” which “behaves like a closure operator.”
How is that? Our first theorem tells us the crucial properties of a closure operator.
120
Theorem 5.18 Suppose Ð\ß g Ñ is a topological space and Eß F are subsets of \ .
1)
2)
3)
4)
5)
Then
clg g œ g
E © clg E
E is closed iff E œ clg E
clg E œ clg Ðclg EÑ
clg ÐE ∪ BÑ œ clg E ∪ clg F
Proof 1) Since g is closed, clg g œ g
2) E © ÖJ À J is closed and E © J × œ clg E
3) Ê : If E is closed, then E is one of the closed sets J used in the definition
clg E œ ∩ ÖJ À J is closed and J ª E×ß so E œ clg E.
É : If E œ clg E, then E is closed because clg E is an intersection of closed sets.
4) clg E is closed, so by 3), clg E œ clg Ðclg EÑ
5) E ∪ F ª E, so clg ÐE ∪ FÑ ª clg E. Likewise, clg ÐE ∪ FÑ ª clg B.
Therefore clg ÐE ∪ F Ñ ª clg ÐEÑ ∪ clg ÐF Ñ.
On the other hand, clg E ∪ clg F is a closed set that contains E ∪ F , and therefore
clg E ∪ clg F ª clg ÐE ∪ F Ñ. ñ
The next theorem tells us that we can use an operator “cl” to create a topology on a set.
Theorem 5.19 (The Closure Operator Theorem) Suppose \ is a set and that for each E © \ ,
a subset cl E is defined ( that is, we have a function cl À c Ð\Ñ Ä c Ð\Ñ ) in such a way that
conditions 1), 2), 4) and 5) of Theorem 5.18 are satisfied. Define g œ {S À
clÐ\  SÑ œ \  S}. Then g is a topology on \ , and cl is now the closure operator for this
topology (that is, cl œ clg ).
Note: i) Such a function cl is called a “Kuratowski closure operator,” or just “closure
operator” for short.
ii) The Closure Operator Theorem does not ask that cl satisfy condition 3): since there
is no topology on the given set \ , 3) would be meaningless. But 3) motivates the definition of g
as the collection of sets whose complements are unchanged when “cl” is applied..
Proof (Numbers in parentheses refer to properties of “cl” )
First note that:
(*)
(5)
if E © F , then cl E © cl E ∪ cl F œ clÐE ∪ FÑ œ cl F
g is a topology on the set \ À
Ð#Ñ
i) \ © cl \ , so \ œ cl \Þ Therefore cl Ð\  gÑ œ cl \ œ \ œ \  g, so g − g .
(1)
Since clÐ\  \Ñ œ cl g œ g œ \  \ , we have that \ − g Þ
ii) Suppose Oα − g for all α − E. For each α! − E, we have
121
(*)
clÐ\  Oα Ñ © clÐ\  Oα! Ñ œ \  Oα! because Oα! − g . This is true for every
α! − A, so clÐ\  Oα Ñ © Ð\  Oα ) œ \  Oα .
But by 2), we know that \  Oα © cl Ð\  Oα Ñ.
Therefore clÐ\  Oα Ñ œ \  Oα , so Oα − g .
iii) If O" and O# are in g , then clÐ\  (O" ∩ O# ÑÑ œ clÐÐ\  O" Ñ ∪ Ð\  O# ÑÑ
(5)
œ clÐ\  O" Ñ ∪ clÐ\  O# Ñ œ Ð\  O" Ñ ∪ Ð\  O# Ñ (since O" and O# − g )
œ \  (O" ∩ O# ). Therefore O" ∩ O# − g . Therefore g is a topology on \ .
With the topology g we now have a closure operator clg . We want to show that clg œ cl.
First, observe that:
(**) clg F œ F iff F is closed in Ð\ß g Ñ iff \  F − g
iff cl Ð\  Ð\  FÑÑ œ \  Ð\  FÑ
iff cl F œ F .
To finish, we must show that cl E œ clg E for every E © \ .
E © clg E, so (*) gives that cl E © clÐclg EÑ. But clg E is closed in Ð\ß g Ñ,
so using F œ clg E in (**) gives cl clg E œ clg E. Therefore cl E © clg E.
(4)
On the other hand, cl cl E œ cl E, so using F œ cl E in (**) gives
that cl E is closed in Ð\ß g ÑÞ
(2)
But cl E ª E, so cl E is one of the closed sets in the intersection that defines
clg E. Therefore cl E ª clg EÞ Therefore cl E œ clg E. ñ
Example 5.20
1) Let \ be a set. For E © \ , define cl E œ 
E if E is finite
.
\ if E is infinite
Then cl satisfies the conditions in the Closure Operator Theorem. Since cl E œ E iff E is finite
or E œ \ , the closed sets in the topology generated by cl are precisely \ and the finite
sets  that is, cl generates the cofinite topology on \ .
2) For each subset E of ‘, define
cl E œ ÖB − ‘ À there is a sequence Ð+8 Ñ in E with each +8 B and |+8  Bl Ä !×.
It is easy to check that cl satisfies the hypotheses of the Closure Operator Theorem. Moreover, a
set E is open in the corresponding topology iff aB − E b,  B such that ÒBß ,Ñ © E. Therefore
the topology generated by cl is the Sorgenfrey topology on ‘. What happens in this example if
“ ” is replaced by “  ” in the definition of cl ?
Since closures, interiors and Frontiers are all related, it shouldn't be surprising that we can also
describe a topology by defining an appropriate “int” operator or “Fr” operator on a set \ .
Exercises
122
E1. Let \ œ Ö!ß "ß #ß Þß Þß Þ× œ Ö!× ∪ . For S © \ , let 9S Ð8Ñ œ the number of elements in
S ∩ Ò"ß 8Ó l œ l S ∩ ["ß 8Ó lÞ Define
g œ Ö S À !  S or Ð! − S and lim 9S8Ð8Ñ œ 1) ×.
8Ä∞
a) Prove that g is a topology on \ .
b) In any space Ð] ß g Ñ : a point B is called a limit point of the set E if
R ∩ ÐE  ÖB×Ñ Á g for every neighborhood R of B. Informally, this means that
B is a limit point of E if there are points of E other than B itself arbitrarily close
to B.)
Prove that in any Ð] ß g Ñ, a subset F is closed iff F contains all of its limit points.
c) For Ð\ß g Ñ as defined above, prove that B is a limit point of \ if and only if B œ !.
E2. Suppose that Eα © Ð\ß g Ñ for each α − E.
a) Suppose that  Öcl Eα À α − E× is closed. Prove that
 Öcl Eα À α − E× œ cl   ÖEα À α − E×
(Note that “ © ” is true for any collection of sets Eα . )
b) A family ÖFα À α − E× of subsets of Ð\ß g Ñ is called locally finite if
each point B − \ has a neighborhood R such that R ∩ Fα Á g for only
finitely many α's. Prove that if ÖFα À α − E× is locally finite, then
 Öcl Fα À α − E× œ cl   ÖFα À α − E×
c) Prove that in Ð\ß g Ñ, the union of a locally finite family of closed sets is closed.
E3. Suppose 0 À Ð\ß .Ñ Ä Ð] ß =ÑÞ Let U be a base for the topology g= and let Æ be a subbase
for g= Þ Prove or disprove: 0 is continuous iff 0 " ÒSÓ is open for all S − U iff 0 " ÒSÓ is open
for all S − Æ Þ
E4.
A space Ð\ß g Ñ is called a X" -space if ÖB× is closed for every B − \Þ
a) Give an example of a space Ð\ß g Ñ with a nontrivial topology and which is not
a X" -space.
b) Prove that \ is a X" -space if and only if, given any two distinct points Bß C − \ , each
point is contained in an open set not containing the other point.
c) Prove that in a X" -space, each set ÖB× can be written as an intersection of open sets.
d) Prove that a subspace of a X" -space is a X" -space.
e) Prove that if a pseudometric space Ð\ß .Ñ is a X" -space, then . must in fact be a
metric.
f) Prove that if \ and ] are X" -spaces, so is \ ‚ ] Þ
E5.
Prove that every infinite X# -space contains an infinite discrete subspace (that is, a subset
which is discrete in the subspace topology).
123
E6.
Suppose that Ð\ß g Ñ and Ð] ß g w Ñ are topological spaces. Recall that the product
topology on \ ‚ ] is the topology for which the collection of “open boxes”
U œ ÖY ‚ Z À Y − g ß Z − g w × is a base.
Therefore a set S © \ ‚ ] is open in the product topology iff for all ÐBß CÑ − S, there exist open
sets Y in \ and Z in ] such that ÐBß CÑ − Y ‚ Z © SÞ (Note that the product topology on
‘ ‚ ‘ is the usual topology on ‘# . We always assume that the “product topology” is the
topology on \ ‚ ] unless something different is explicitly stated. )
a) Verify that U is, in fact, a base for a topology on \ ‚ ] .
b) Prove that the projection map 1B À \ ‚ ] Ä \ is an open map.
c) Prove that if E © \ and F © ] , then cl\‚] ÐE ‚ FÑ œ cl\ E ‚ cl] FÞ Use this to
explain why “the product of two closed sets is closed in \ ‚ ] .”
d) Show that \ and ] each has a countable base iff \ ‚ ] has a countable base. Show
that there is a countable neighborhood base at ÐBß CÑ − \ ‚ ] iff there is a countable
neighborhood base UB at B − \ and a countable neighborhood base UC at C − ] Þ
e) Suppose Ð\ß ." Ñ and Ð] ß .# Ñ are pseudometric spaces. Define a pseudometric . on
the set \ ‚ ] by
.ÐÐB" ß C" Ñß ÐB# ß C# ÑÑ œ ." ÐB" ß B# Ñ  .# ÐC" ß C# ÑÞ
Prove that the product topology on \ ‚ ] is the same as the topology g. .
Note: . is the analogue of the taxicab metric in ‘# Þ Of course, there are other
equivalent pseudometrics producing the product topology on \ ‚ ] , e.g.,
. w ÐÐB" ß C" Ñß ÐB# ß C# ÑÑ œ Ð." ÐB" ß B# Ñ#  .# ÐC" ß C# Ñ# Ñ"Î# ß or
. ww ÐÐB" ß C" Ñß ÐB# ß C# ÑÑ œ maxÖ." ÐB" ß B# Ñß .# ÐC" ß C# Ñ×Þ
E7.
Ð\ß g Ñ is called a X# -space ( or Hausdorff space ) if whenever Bß C − \ and B Á C, then
there exist disjoint open sets Y and Z with B − Y and C − Z .
a) Give an example of a space Ð\ß g Ñ which is a X" -space but not a X# -space.
b) Prove that a subspace of a Hausdorff space is Hausdorff.
c) Prove that if \ and ] are Hausdorff, then so is \ ‚ ] .
124
E8.
Suppose E © Ð\ß g ÑÞ The set E is called regular open if E œ int Ðcl EÑ and E is
regular closed if E œ cl (int E).
a) Show that for any subset F
i) \  cl F œ int Ð\  FÑ
ii) \  int F œ cl Ð\  FÑ
b) Give an example of a closed subset of ‘ which is not regular closed.
c) Show that the complement of a regular open set in Ð\ß g Ñ is regular closed
and vice-versa.
d) Show that the interior of any closed set in Ð\ß g Ñ is regular open.
e) Show that the intersection of two regular open sets in Ð\ß g Ñ is regular open.
f) Give an example of the union of two regular open sets that is not regular open.
E9.
Prove or give a counterexample:
a) For any B in Ð\ß g Ñß ÖB× is equal to the intersection of all
open sets containing B.
b) In Ð\ß g Ñ, a finite set must be closed.
c) If for each α − E, each gα is a topology on \ , then {gα : α − E} is a topology
on \ .
d) If g" and g# are topologies on X, then there is a unique smallest topology g$ on X
such that g$ ª g" ∪ g# .
e) Suppose, for each α − E, that gα is a topology on \ . Then there is a unique smallest
topology g on \ such that for each α, g ª gα Þ
E10. Assume that each integer (except is divisible by a prime but nothing else about prime
numbers. (This assumption is equivalent to assuming that each natural number bigger than can be
factored into primes). For + − ™ and . − ß let
and let
F+ß. œ ÖÞÞÞß +  #.ß +  .ß +ß +  .ß +  #.ß ÞÞÞ× œ Ö+  5. À 5 − ™×
U œ ÖF+ß. À + − ™ß . − × Ð=9 U is the set of all arithmetic progressions in ™Ñ
a) Prove that U is a base for a topology g on ™.
b) Show that each set is closed in (™,g )Þ
c) What is the set  ÖF!ß: À : a prime number×? Explain why this set is not
closed in Ð™ß g Ñ.
d) What does part c) tell you about the set of prime numbers?
125
6. Countability Properties of Spaces
Countable sets are generally easier to work with than uncountable sets, so it is not surprising that
spaces with certain “countability properties” are viewed as desirable. Most of these properties
have already been defined, but the definitions are collected together here for convenience.
Definition 6.1 Ð\ß g Ñ is called
first countable (or, is said to satisfy the first axiom of countability) if we can choose a
countable neighborhood base UB at every point B − \
second countable (or, is said to satisfy the second axiom of countability) if there
is a countable base U for the topology g
separable if there is a countable dense set H in \
Lindelöf if whenever h is a collection of open sets for which h œ \ , then there is
a countable h w œ ÖY" ß Y# ß ÞÞÞß Y8 ß ÞÞÞ× © h for which h w œ \Þ
h is called an open cover of \ , and h w is called a subcover from h . Thus, \ is
Lindelof
¨ if “every open cover has a countable subcover.”)
Example 6.2
1) A countable discrete space Ð\ß g Ñ is second countable because U œ ÖÖB× À B − \× is
a countable base.
2) ‘ is second countable because U œ ÖÐ+ß ,Ñ À +ß , − × is a countable base. Similarly,
‘8 is second countable since the collection U of boxes ÐB" ß C" Ñ ‚ ÐB# ß C# Ñ ‚ ÞÞÞ ‚ ÐB8 ß C8 Ñ with
rational endpoints is a countable base (check!)
3) Let \ be a countable set with the cofinite topology g . \ has only countably many
finite subsets (see Theorem I.11.1) so there are only countably many sets in g . Ð\ß g Ñ is second
countable because we could choose U œ g as a countable base.
The following theorem implies that each of the spaces in the preceding example is also first
countable, separable, and Lindelof.
¨ (However, it is really worthwhile to try to verify each of those
assertions directly from the definitions.)
Theorem 6.3 A second countable topological space (\ß g Ñ is also first countable, separable, and
Lindelof.
¨
Proof Let U œ ÖS" ß S# ß ÞÞÞS8 ß ÞÞÞ × be a countable base for g .
i) For each 8, pick a point B8 − S8 and let H œ ÖB8 À 8 − ×. The countable set H is
dense. To see this, notice that if Y is any nonempty open set in \ , then for some 8,
B8 − S8 © Y so Y ∩ H Á g. So \ is separable.
126
ii) For each B − \ß let UB œ {S − U À B − S×. Clearly UB is a neighborhood base at B,
so \ is first countable.
iii) Let h be any open cover of \ . If B − \ , then B − some set YB − h Þ For each B, we
can then pick a basic open set SB − U such that B − SB © YB . Let i œ ÖSB À B − \×Þ Since
each SB − U ß there can be only countably many different sets SB : that is, i may contain
“repeats.” Eliminate repeats and list only the different sets in i , so i œ ÖSB" ß SB# ß ÞÞÞSB8 ß ÞÞÞ ×
where SB8 © YB8 − h Þ Every B is in one of the sets SB8 , so h w œ ÖYB" ß YB# ß ÞÞÞYB8 ß ÞÞÞ× is a
countable subcover from h . Therefore \ is Lindelof.
¨ ñ
The following examples show that no other implications exist among the countability properties
mentioned in Theorem 6.3.
Example 6.4
1) Suppose \ is uncountable and let g be the cofinite topology on \ .
Ð\ß g Ñ is separable since any infinite set is dense.
Ð\ß g Ñ is Lindelof.
¨ To see thisß let h be an open cover of \ . Pick any one
nonempty set Y − h . Then \  Y is finite, say \  Y œ ÖB" ß ÞÞÞß B8 ×Þ For
each B3 ß pick a set Y3 − h with B3 − Y3 Þ Then h w œ ÖY × ∪ ÖY3 À 3 œ "ß ÞÞÞß 8×
is a countable (actually, finite) subcover chosen from h .
However \ is not first countable (Example 4.5.3), so Theorem 6.3, \ is also not
second countable.
2) Suppose \ is uncountable. Define g œ ÖS © \ À S œ g or \  S is countable×.
g is a topology on \ (check! ) called the cocountable topology. A set G © \ is
closed iff G œ \ or G is countable. (This is an “upscale” analogue of the
cofinite topology.)
An argument very similar to the one in the preceding example shows that Ð\ß g Ñ
is Lindelof.
¨ But Ð\ß g Ñ is not separable  every countable subset is closed
and therefore not dense. By Theorem 6.3, Ð\ß g Ñ also cannot be second
countable.
3) Suppose \ is uncountable set and choose a particular point : − \Þ
Define g œ ÖS © \ À S œ g or : − S×. (Check that g is a topology.)
Ð\ß g Ñ is separable because Ö:× is dense.
Ð\ß g Ñ is not Lindelof
¨  because the cover h œ ÖÖBß :× À B − \× has no
countable subcover.
Is Ð\ß g Ñ first countable?
4) Suppose \ is uncountable and let g be the discrete topology on \ .
127
Ð\ß g Ñ is first countable because UB œ ÖÖB×× is a neighborhood base at B
Ð\ß g Ñ is not second countable because each open set ÖB× would have to
be in any base U .
Ð\ß g Ñ is not separable: for every countable set H, cl H œ H Á \Þ
Ð\ß g Ñ is not Lindelof
¨ because the cover h œ ÖÖB× À B − \× has no countable
subcover. (In fact, not a single set can be omitted from h : h has no proper
subcovers at all. )
For “special” topological spaces  pseudometrizable ones, for example  it turns out that things
are better behaved. For example, we noted earlier that every pseudometric space Ð\ß .Ñ is first
countable. The following theorem shows that in Ð\ß .Ñ the other three countability properties are
equivalent to each other: that is, either all of them are true in Ð\ß .Ñ or none are true.
Theorem 6.5 Any pseudometric space Ð\ß .Ñ is first countable. Ð\ß .Ñ is second countable iff
Ð\ß .Ñ is separable iff Ð\ß .Ñ is Lindelof.
¨
Proof i) Second countable Ê Lindelof:
¨ by Theorem 6,3, this implication is true in any
topological space.
ii) Lindelof
For each 8 − , let
¨ Ê separable: suppose Ð\ß .Ñ is Lindelof.
¨
h8 œ ÖF " ÐBÑ À 8 − ß B − \×. For each 8, h8 is an open cover so h8 has a countable
8
subcover  that is, for each 8 we can find countably many 8" -balls that cover \ À say
\ œ ∞
5œ" F 8" ÐB8ß5 ÑÞ Let H be the set of centers of all these balls: H œ ÖB8ß5 À 8ß 5 − ×Þ For
any B − \ and every 8, we have B − F " ÐB8ß5 Ñ for some 5 , so .ÐBß B8ß5 Ñ  8"  so B can be
8
approximated arbitrarily closely by points from H. Therefore H is dense, so Ð\ß .Ñ is separable.
iii) Separable Ê second countable: suppose Ð\ß .Ñ is separable and that
H œ ÖB" ß B# ß ÞÞÞß B5 ß ÞÞÞ× is a countable dense set. Let U œ ÖF " ÐB5 Ñ À 8ß 5 − ×. U is a countable
8
collection of open balls and we claim U is a base for the topology g. Þ
Suppose C − S − X. Þ By the definition of “open,” we have F% ÐCÑ © S
for some %  !. Pick 8 so that 8"  #% ß and pick B5 − H so that .ÐB5 ß CÑ  8" Þ Then
C − F " ÐB5 Ñ © F% ÐCÑ © S (because D − F " ÐB5 Ñ Ê .ÐDß CÑ Ÿ .ÐDß B5 Ñ  .ÐB5 ß CÑ
8
8
 8"  8"  #Ð #% Ñ œ % ). ñ
It's customary to call a metric space that has these three equivalent properties a “separable
metric space” rather than a “second countable metric space” or “Lindelof
¨ metric space.”
Theorem 6.5 implies that the spaces in parts 1), 2), 3) of Example 6.4 are not pseudometrizable.
In general, to show a space Ð\ß g Ñ is not pseudometrizable we can i) show that it fails to have
some property common to all pseudometric spaces (for example, first countability), or ii) show
that it has one but not all of the properties “second countable,” “Lindelof,”
¨ or “separable.”
128
Exercises
E11.
Define g œ ÖY ∪ Z À Y is open in the usual topology on ‘ and Z © ×.
a) Show that g is a topology on ‘. If B is irrational, describe an “efficient”
neighborhood base at B . Do the same if B is rational.
b) Is Ð‘ß g Ñ first countable? second countable? Lindelof
¨ ? separable?
The spaceÐ‘ß g Ñ is called the “scattered line.” We could change the definition of g by
replacing  with some other set E © ‘ß creating a topology in which the set E is “scattered.”
Hint: See Example I.7.9.6. It is possible to find open intervals M8 such that ∞
8œ" M8 ª 
and for which  lengthÐM8 Ñ  ".
∞
8œ"
E12. A point B − Ð\ß g Ñ is called a condensation point if every neighborhood of B is
uncountable.
a) Let G be the set of all condensation points in \Þ Prove that G is closed.
b) Prove that if \ is second countable, then \  G is countable.
E13. Suppose Ð\ß g Ñ is a second countable space and let U be a countable base for the
topology. Suppose U w is another base (not necessarily countable ) for g containing open sets all
of which have some property T . (For example, “T ” could be “clopen” or “separable.”) Show
that there is a countable base U ww consisting of open sets with property T Þ
Hint: think about the Lindelof
¨ property. )
E14.
A space Ð\ß g Ñ is called hereditarily Lindelof
¨Þ
¨ if every subspace of \ is Lindelof
a) Prove that a second countable space is hereditarily Lindelof.
¨
In any space, a point B is called a limit point of the set E if R ∩ ÐE  ÖB×Ñ Á g for every
neighborhood R of BÞ Informally, B is a limit point of E if there are points in E different
from B and arbitrarily close to B.)
b) Suppose \ is hereditarily Lindelof
¨Þ
Prove that E œ ÖB − \ À B is not a limit point of \× is countable.
E15. A space Ð\ß g Ñ is said to satisfy the countable chain condition ( œ CCC Ñ if every family
of disjoint open sets must be countable.
a) Prove that a separable space Ð\ß g Ñ satisfies the CCC.
b) Give an example of a space that satisfies the CCC but that is not separable. (It is not
necessary to do so, but can you find an example which is a metric space?)
E16.
Suppose c and U are two bases for the topology in Ð\ß g Ñ, and that c and U
129
are infinite.
a) Prove that there is a subfamily U w © U such that U w is also a base and l U w l Ÿ lc l.
(Hint: For each pair > œ ÐY ß Z Ñ − c ‚ c , pick, if possible, a set [> − U such that
Y © [ © Z ; otherwise set [> œ gÞ )
b) Use part a) to prove that the Sorgenfrey line is not second countable.
(Hint: Show that otherwise there would be a countable base of sets of the form Ò+ß ,Ñß but
that this is impossible. Ñ
130
7. More About Subspaces
Suppose E © \ß where Ð\ß g Ñ is a topological space. We defined the subspace topology gE on
E in Definition 3.1: gE œ ÖE ∩ S À S − g × .
In this section we explore some simple but important properties of subspaces.
If E © F © \ , there are two ways to put a topology on E À
1) we can give E the subspace topology gE from \ , or
2) we can give F the subspace topology gF , and then give E the subspace topology from
the space ÐFß gF Ñ  that is, we can give E the topology ÐgF ÑE Þ
In other words, we can think of E as a subspace of \ or as a subspace of the subspace F.
Fortunately, the next theorem says that these two topologies are the same. More informally,
Theorem 7.1 says that “a subspace of a subspace is a subspace.”
Theorem 7.1 If E © F © \ , and g is a topology on \ , then gE œ (gF )E .
Proof Y − gE iff Y œ S ∩ E for some S − g iff Y œ S ∩ ÐF ∩ EÑ iff Y œ ÐS ∩ FÑ ∩ E.
But S ∩ F − gF , so the last equation holds iff S − ÐgF ÑE . ñ
We always assume a subset E has the subspace topology (unless something else is explicitly
stated). The notation E © Ð\ß g Ñ emphasizes that E is considered a subspace, not merely a
subset E © \Þ
By definition, a set is open in the subspace topology on E iff it is the intersection with E of an
open set in \ . We can prove the same is also true for closed sets.
Theorem 7.2 Suppose E © Ð\ß g Ñ. J is closed in E iff J œ E ∩ G where G is closed in \ .
Proof J is closed in E iff E  J is open in E iff E  J œ S ∩ E (for some open set S in \ )
iff J œ E  ÐS ∩ EÑ œ Ð\  SÑ ∩ E œ G ∩ E (where G œ \  S is a closed set in \ ). ñ
Theorem 7.3 Suppose E © Ð\ß g Ñ.
1) Let a − E. If U+ is a neighborhood base at a in \ , then ÖF ∩ E À F − U+ × is a
neighborhood base at + in EÞ
2) If U is a base for g , then ÖF ∩ E À F − U × is a base for gE Þ
Slightly abusing notation, we can informally write these collections as U+ ∩ E and.U ∩ EÞ Why
is this an “abuse?” What do U+ ∩ E and U ∩ E mean if taken literally?
131
Proof 1) Suppose + − E and that R is a neighborhood of + in E. Then + − intE R © R , so
there is an open set S in \ such that + − intE R œ S ∩ E © SÞ
Since U+ is a neighborhood base at + in \ , there is a neighborhood F − U+ such that
+ − F © SÞ Then + − Ðint\ FÑ ∩ E © F ∩ E − U+ ∩ E. Since Ðint\ FÑ ∩ E is open in E, we see
that F ∩ E is a neighborhood of + in E. And since + − F ∩ E © S ∩ E œ intE R © R , we see
that U+ ∩ E is a neighborhood base at + in E. ñ
2) Exercise
Theorem 7.3 tells us that we can get a neighborhood base at a point + − E by choosing a
neighborhood base at + in \ are then restricting all its sets to E; and that the same applies to a
base for the subspace topology.
Corollary 7.4 Every subspace of a first countable (or second countable) space Ð\ß g Ñ is first
countable (or second countable).
Example 7.5 Suppose W " is a circle in ‘# and that : − W " © ‘# Þ U: œ ÖF% Ð:Ñ À %  !× is a
neighborhood base at : in ‘# , and therefore U: ∩ W " is a neighborhood base at : in the subspace
W " . The sets in U: ∩ W " are “open arcs on W " containing :.” (See the figure,)
The following theorem relates closures in subspaces to closures in the larger space. It is a useful
technical tool.
Theorem 7.6 Suppose E © F © Ð\ß g Ñ, then clF E œ F ∩ cl\ EÞ
Proof F ∩ cl\ E is a closed set in F that contains Eß so F ∩ cl\ E ª clF EÞ
On the other hand, suppose , − F ∩ cl\ E. To show that , − clF E, pick an open set Y in
F that contains ,. We need to show Y ∩ E Á g. There is an open set S in \ such that
S ∩ F œ Y . Since , − cl\ E, we have that g Á S ∩ E œ S ∩ ÐF ∩ EÑ œ ÐS ∩ FÑ ∩ E
œ Y ∩ E. ñ
132
Example 7.7
1)  œ cl  œ cl‘  ∩ 
2) clÐ!ß#Ó Ð!ß "Ñ œ cl‘ Ð!ß "Ñ ∩ Ð!ß #Ó œ Ò!ß "Ó ∩ Ð!ß #Ó œ Ð!ß "Ó
3) The analogous results are not true for interiors and boundaries. For example:
 œ int  Á int‘  ∩  œ g , and
g œ Fr  Á Fr‘  ∩  œ Þ
Why does “cl” have a privileged role here? Is there a “reason” why you would expect a better
connection between closures in E and closures in \ than you would expect between interiors in
E and interiors in \ ?
Definition 7.8 A property T of topological spaces is called hereditary if whenever a space \
has property T , then every subspace E also has property T .
For example, Corollary 7.4 tells us that first and second countability are hereditary properties.
Other hereditary properties include “finite cardinality” and “pseudometrizability.” On the other
hand, “infinite cardinality” is not a hereditary property.
Example 7.9
1) Separability is not a hereditary property. For example, consider the Sorgenfrey plane
\ Ðsee Example 5.4)Þ \ is separable because # is dense.
Consider the subspace H œ ÖÐBß CÑ À B  C œ "×. The set Y œ Ò+ß +  "Ñ ‚ Ò,ß ,  "Ñ
is open in \ so if Ð+ß ,Ñ − Hß then Y ∩ H œ ÖÐ+ß ,Ñ× is open in the subspace HÞ Therefore H is
a discrete subspace of \ , and an uncountable discrete space is not separable.
Similarly, the Moore place > is separable (see Example 5.6); the B-axis in > is an
uncountable discrete subspace which is not separable.
2) The Lindelof
¨ property is not heredity. Let E be an uncountable set and let : be an
additional point not in E. Define \ œ E ∪ Ö:×Þ Put a topology on \ by describing a
neighborhood base at each point.
U+ œ ÖÖ+×× for + − E
 U œ ÖF À : − F and \  F is finite×
:
(Check that the collections UB satisfy the hypotheses of the Neighborhood Base Theorem 5.2.)
If i is an open cover of \ , then : − Z for some Z − i . By ii), every neighborhood of : in \
has a finite complement, so \  Z is finite. For each C in the finite set \  Z , we can choose a
set ZC − i with C − ZC Þ Then i w œ ÖZ × ∪ ÖZC À C − \  Z × is a countable (in fact, finite)
subcover from i , so \ is Lindelof.
¨
The definition of U+ implies that each point of E is isolated in E; that is, E is an uncountable
discrete subspace and h œ ÖÖ+× À + − E× is an open cover of E that has no countable subcover.
Therefore the subspace E is not Lindelof.
¨
133
Even if a property is not hereditary, it is sometimes “inherited” by certain subspaces as the next
theorem illustrates.
Theorem 7.10 A closed subspace of a Lindelof
¨ space is Lindelof
¨
property is “closed hereditary.” )
(We say that the Lindelof
¨
Proof Suppose O is a closed subspace of the Lindelof
¨ space \ . Let h œ ÖYα À α − E× be a
cover of O by sets Yα that are open in O . For each α, there is an open set Zα in \ such that
Z α ∩ O œ Yα Þ
Since O is closed, \  O is open and i œ ÖÖ\  O×× ∪ ÖZα À α − E× is an open cover of \Þ
But \ is Lindelof,
¨ so i has a countable subcover from i , say i w œ Ö\  Oß Z" ß ÞÞÞß Z8 ß ÞÞÞ×Þ
(The set \  O may not be needed in i w , but it can't hurt to include it.) Clearly then, the
collection ÖY" ß ÞÞÞß Y8 ß ÞÞÞ× is a countable subcover of O from h . ñ
(A little reflection on the proof shows that to prove O is Lindelof,
¨ it would be equivalent to
show that every cover of O by sets open in \ has a countable subcoverÞÑ
134
8. Continuity
To define continuous functions between pseudometric spaces, we began by using the distance
function . . But then we proved that our definition is equivalent to other formulations stated in
terms of open sets, closed sets, or neighborhoods. In the context of topological spaces we do not
have distance functions to describe “nearness.” But we can still use neighborhoods of + to take
about “nearness” to +.
Definition 8.1 A function 0 À Ð\ß g Ñ Ä Ð] ß g w Ñ is continuous at + − \ if whenever R is a
neighborhood of 0 Ð+Ñ, then 0 " ÒR Ó is a neighborhood of +. We say 0 is continuous if 0 is
continuous at each point of \ .
The statement that 0 is continuous at + is clearly equivalent to each of the following statements: :
i) for each neighborhood R of 0 Ð+Ñ there is a neighborhood [ of + such
that 0 Ò[ Ó © R
ii) for each open set Z containing 0 Ð+Ñ there is an open set Y containing + such
that 0 ÒY Ó © Z
iii) for each basic open set Z containing 0 Ð+Ñ there is a basic open set Y
containing + such that 0 ÒY Ó © Z .
The conditions i) - iii) for continuity in the following theorem are the same as those in Theorem
II.5.6 for pseudometric spaces. Condition iv) was not mentioned in Chapter II, but it is
sometimes handy.
Theorem 8.2 Suppose 0 À Ð\ß g Ñ Ä Ð] ß g w Ñ. The following are equivalent.
i) 0 is continuous
ii) if S − g w ß then 0 " ÒSÓ − g (the inverse image of an open set is open)
iii) if J is closed in ] , then 0 " ÒJ Ó is closed in \ (the inverse image of a
closed set is closed)
iv) for every E © \ À 0 Òcl\ ÐEÑÓ © cl] Ð0 ÒEÓÑ.
Proof The proof that i) - iii) are equivalent is identical to the proof for Theorem II.5.6 for
pseudometric spaces. That proof was deliberately worded in terms of open sets, closed sets, and
neighborhoods so that it would carry over to this new situation.
iiiÑ Ê iv) 0 " Òcl] Ð0 ÒEÓÑÓ ª 0 " Ò0 ÒEÓÓ ª E. Since cl] Ð0 ÒEÓÑ is closed in ] ß iii) gives
that 0 Òcl] Ð0 ÒEÓÑÓ is a closed set in \ that contains EÞ Therefore 0 " Òcl] Ð0 ÒEÓÑÓ ª cl\ ÐEÑ, so
Òcl] Ð0 ÒEÓÑÓÓ ª 0 Òcl\ ÐEÑÓÞ
"
ivÑ Ê i) Suppose + − \ and that R is a neighborhood of 0 Ð+ÑÞ Let O œ \  0 " ÒR Ó
and Y œ \  cl\ O © 0 " ÒR ÓÞ Y is open, and we claim that + − Y  which will show
that 0 " ÒR Ó is a neighborhood of +, completing the proof. So we need to show that
+ Â cl\ O . But this is clear: if + − cl\ O , we would have
0 Ð+Ñ − 0 Òcl\ O Ó © cl] 0 ÒOÓ, which is impossible since R ∩ 0 ÒOÓ œ gÞ ñ
135
Example 8.3 Sometimes we want to know whether a certain property is “preserved by
continuous functions”  that is, if \ has property T and 0 À \ Ä ] is continuous and onto,
must ] also have the property T ?
Condition iv) in Theorem 8.2 implies that continuous maps preserve separability.
Suppose H is a countable dense set in \ . Then 0 ÒHÓ is countable and 0 ÒHÓ is dense in ] because
] œ 0 Ò\Ó œ 0 Òcl HÓ © cl 0 ÒHÓÞ
By contrast, continuous maps do not preserve first countability: for example, let Ð] ß g Ñ
be any topological space. Let g w be the discrete topology on ] . Ð] ß g w Ñ is first countable and
the identity map 3 À Ð] ß g w Ñ Ä Ð] ß g Ñ is continuous and onto. Thus, every space Ð] ß g Ñ is the
continuous image of a first countable space.
Do continuous maps preserve other properties  such as Lindelof,
¨ second countable, or
metrizable?
The following theorem makes a few simple and useful observations about continuity.
Theorem 8.4 Suppose 0 À Ð\ß g Ñ Ä Ð] ß g w Ñ.
1) Let F œ ranÐ0 Ñ © ] Þ Then 0 is continuous iff 0 À Ð\ß g Ñ Ä ÐFß gFw Ñ is continuous.
In other words, F œ the range of f Ða subspace of the codomain ] Ñ is what matters
for the continuity of 0 Þ Points (if any) in ]  F are irrelevant. For example, the
function sin À ‘ Ä ‘ is continuous iff the function sin À ‘ Ä Ò  "ß "Ó is continuous.
2) Let E © \ . If 0 is continuous, then 0 lE œ 1 À E Ä ] is continuousÞ
That is, the restriction of a continuous function to a subspace is continuous.
For example, sin À  Ä Ò  "ß "Ó is continuous.)
3) If f is a subbase for g w (in particular, if f is a base), then 0 is continuous iff 0 " ÒY Ó
is open whenever Y − f .
To check continuity, it is sufficient to show that the inverse image of every =?,basic
open set is open.
Proof 1) Exercise: the crucial observation is that if S © ] , then 0 " ÒSÓ œ 0 " ÒF ∩ SÓ.
2) If S is open in ] , then 1" ÒSÓ œ 0 " ÒSÓ ∩ E  which is an open set in E.
3) Exercise: it depends only on the definition of a subbase and set theory:
0 " ÒYα À α − EÓ œ Ö0 " ÒYα Ó À α − E× and
0 " ÒYα À α − EÓ œ Ö0 " ÒYα Ó À α − E× ñ
Example 8.5
1) If \ has the discrete topology and ] is any topological space, then every function
0 À \ Ä ] is continuous.
2) Suppose \ has the trivial topology and that 0 À \ Ä ‘. If 0 is constant, then 0 is
continuous. If 0 is not constant, then there are points +ß , − \ for which 0 Ð+Ñ Á 0 Ð,ÑÞ
Let M be an open set in ‘ containing 0 Ð+Ñ but not 0 Ð,ÑÞ Then 0 " ÒMÓ is not open in \
so 0 is not continuous. We conclude that 0 is continuous iff 0 is constant.
(In this example, we could replace ‘ by any metric space Ð] ß .Ñß or by any
topological space Ð] ß g Ñ that has what property?
3) Let \ be a rectangle inscribed inside a circle ] centered at T . For + − \ , let
136
0 Ð+Ñ be the point where the ray from T through + intersects ] . (The function 0 is called
a “central projection.” ). Then both 0 À \ Ä ] and 0 " À ] Ä \ are continuous
bijections.
Example 8.6 (Weak topologies) Suppose \ is a set. Let Y œ Ö0α À α − E× be a collection of
functions where each 0α À \ Ä ‘Þ If we put the discrete topology on \ , then all of the function
0α will be continuous. But a topology on \ smaller than the discrete topology might also make
all the 0α 's continuous. The smallest topology on \ that makes all the 0α 's continiuous is called
the weak topology g on \ generated by the collection Y .
How can we describe that topology more directly? g makes all the 0α 's continuous iff for each
open S © ‘ and each α − E, the set 0α" ÒSÓ is in g . Therefore the weak topology generated by
Y is the smallest topology that contains all these sets. According to Example 5.17.5, this means
that weak topology g is the one for which the collection Æ œ Ö0α" ÒSÓ À S open in ‘, α − E× is
a subbase. (It is clearly sufficient here to use only basic open sets S from ‘  that is, open
intervals Ð+ß ,Ñ À why ? Would using all open sets S put any additional sets into g ? )
For example, suppose \ œ ‘# and that Y œ Ö1B ß 1C × contains the two projection maps
1B ÐBß CÑ œ B and 1C ÐBß CÑ œ C. For an open interval Y œ Ð+ß ,Ñ © ‘, 1B" ÒY Ó is the “open
vertical strip” Y ‚ ‘; and 1C" ÒZ Ó is the “open horizontal strip” ‘ ‚ Z . Therefore a subbase for
the weak topology on ‘# generated by Y consists of all such open horizontal or vertical strips.
Two such strips intersect in an “open box” Ð+ß ,Ñ ‚ Ð-ß .Ñ in ‘# , so it is easy to see that the weak
topology is the product topology on ‘ ‚ ‘, that is, the usuall topology of ‘# .
Suppose E © ‘ and that 3 À E Ä ‘ is the identity function 3ÐBÑ œ BÞ What is the weak topology
on the domain E generated by the collection Y œ Ö3× ?
137
Definition 8.7 0 À Ð\ß g Ñ Ä Ð] ß g w Ñ is called open if whenever S is open in \ , then 0 ÒSÓ is
open in ] , and 0 is called closed if 0 ÒJ Ó is closed in ] whenever J is closed in \ .
Suppose l\l  ". Let g be the discrete topology on \ and let g w be the trivial topology on \ .
The identity map 3 À Ð\ß g Ñ Ä Ð\ß g w Ñ is continuous but neither open nor closed,
and 3 À Ð\ß g w Ñ Ä Ð\ß g Ñ is both open and closed but not continuous. Open (closed) maps are
quite different from continuous maps  even if 0 is a bijection! Here are some examples that are
more interesting.
Example 8.8
1) 0 À Ò!ß #1Ñ Ä W " œ ÖÐBß CÑ − ‘# À B#  C# œ "× given by 0 Ð)Ñ œ Ðcos ),sin )Ñ.
It is easy to check that 0 is continuous, one-to-one, and onto. J œ Ò1ß #1Ñ is closed in Ò!ß #1Ñ but
0 Ò Ò1ß #1ÑÓ is not closed in W " . Also, Ò!ß 1Ñ is open in Ò!ß #1Ñ but 0 Ò Ò!ß 1Ñ Ó Ó is not open in W " .
A continuous, one-to-one, onto mapping does not need to be open or closed x
2) Suppose \ and ] are topological spaces and that 0 À \ Ä ] is a bijection so there is
an inverse function 0 " À ] Ä \Þ Then 0 " is continuous iff 0 is open. To check this, let
1 œ 0 " Þ For an open set S © \ß we have C − 1" ÒSÓ iff 1ÐCÑ − S iff C œ 0 Ð1ÐCÑÑ − 0 ÒSÓ , so
0 ÒSÓ œ 1" ÒSÓ. So 0 ÒSÓ is open iff 1" ÒSÓ is open. For a bijection 0 , “0 is open” is equivalent to
“0 " is continuous.”
If S is replaced by a closed set J © \ , then a similar argument also shows that a
bijection 0 is closed iff 0 " is continuous.
Definition 8.9 A mapping 0 À Ð\ß g Ñ Ä Ð] ß g w Ñ is called a homeomorphism if 0 is a bijection
and 0 and 0 " are both continuous. If a homeomorphism 0 exists, we say that \ and ] are
homeomorphic and write \ ¶ ] .
Note: The term is “homeomorphism," not “homomorphism” (a term from algebra)Þ The
etymologies are closely related: “-morphism” comes from the Greek word Ð.939(Ñ for
“shape” or “form.” The prefixes “homo” and “homeo” come from Greek words
meaning “same” and “similar” respectively. There was a major dispute in western
religious history, mostly during the 4>2 century AD, that hinged on the distinction between
“homeo” and “homo.”
As noted in the preceding example, we could also describe a homeomorphism as a continuous
open bijection or a continuous closed bijection.
It is obvious that in a collection of topological spaces, the homeomorphism relation “ ¶ ” is an
equivalence relation: if \ß ] ß and ^ are topological spaces, then
i) \ ¶ \
ii) if \ ¶ ] , then ] ¶ \
iii) if \ ¶ ] and ] ¶ ^ , then \ ¶ ^ .
138
Example 8.10
1) The function 0 À Ò!ß #1Ñ Ä W " given by 0 Ð)Ñ œ Ðcos ),sin )Ñ is not a
homeomorphism even though 0 is continuous, 1  1, and onto.
2) The “central projection” from the rectangle to the circle (Example 8.5.3) is a
homeomorphism.
3) Using a linear map, it is easy to see that any two open intervals Ð+ß ,Ñ in ‘ are
homeomorphic. The mapping tan À Ð  1# ß 1# Ñ Ä ‘ is a homeomorphism, so that each
nonempty open interval in ‘ is actually homeomorphic to ‘ itself.
3) If 0 À Ð\ß .Ñ Ä Ð] ß =Ñ is an isometry (onto) between metric spaces, then both 0 and
0 " are continuous, so 0 is a homeomorphism.
4) If . and . w are equivalent metrics (so g. œ g. w Ñ, then the identity map
3 À Ð\ß .Ñ Ä Ð\ß . w Ñ is a homeomorphism. A homeomorphism 0 between metric spaces
need not preserve distances, that is, 0 need not be an isometry.
5) The function 0 À Ö 8" À 8 − × Ä  given by 0 Ð 8" Ñ œ 8 is a homeomorphism Ðboth
spaces have the discrete topology! ) Topologically, these spaces are the identical: both
are just countable infinite sets with the discrete topology. 0 is not an isometry
Any two discrete spaces \ and ] with the same cardinality are homeomorphic: any
bijection between them is a homeomorphism.
139
6) Let T denote the “north pole” of the sphere
W # œ ÖÐBß Cß DÑ À B#  C#  D # œ "× © ‘$ Þ
The function 0 illustrated below is a “stereographic projection”. The arrow starts at T , runs
inside the sphere and exits through the surface of the sphere at a point ÐBß Cß DÑ. 0 ÐBß Cß DÑ is the
point where the tip of the arrow hits the BC-plane ‘# . Thus 0 maps each point in W #  ÖT × to a
point in ‘# . The function 0 is a homeomorphism. (See the figure below.)
What is the significance of a homeomorphism 0 À \ Ä ] ?
i) 0 is a bijection so it sets up a perfect one-to-one correspondence between the points in
\ and ] : B Ç C œ 0 ÐBÑ. We can imagine that 0 just “renames” the points in \ . There is also
a perfect one-to-one correspondence between the subsets \α and ]α of \ and ] :
\α Ç ]α œ 0 Ò\α Ó. Because 0 is a bijection, each subset ]α © ] corresponds in this way to
one and only one subset \α © \ .
ii) 0 is a bijection, so 0 treats unions, intersections and complements “nicely”:
a)
b)
c)
d)
e)
0 " Ò]α À α − EÓ œ Ö0 " Ò]α Ó À α − E×
0 " Ò]α À α − EÓ œ Ö0 " Ò]α Ó À α − E×, and
0 Ò\α À α − EÓ œ Ö0 Ò\α Ó À α − E×
0 Ò\α À α − EÓ œ Ö0 Ò\α Ó À α − E×ß and
0 Ò\  GÓ œ 0 Ò\Ó  0 ÒGÓ œ ]  0 ÒGÓ
(Actually a),b), c) are true for any function; but d) and e) depend on 0 being a bijection.)
These properties say that this correspondence between subsets preserves unions: if each
\α Ç ]α ß then \α Ç ]α œ 0 Ò\α Ó œ 0 Ò\α Ó. Similarly, 0 preserves intersection and
complements.
140
iii) Finally if 0 and 0 " are continuous, then open (closed) sets in \ correspond to
open(closed) sets in ] and vice-versa.
The total effect is all the “topological structure” in \ is exactly “duplicated” in ] and vice versa:
we can think of points, subsets, open sets and closed sets in ] are just “renamed copies” of their
counterparts in \ . Moreover 0 preserves unions, intersections and complements, so 0 also
preserves all properties of \ that can defined be using unions, intersections and complements of
open sets. For example, we can check that if 0 is a homeomorphism and E © \ , then
0 Òint\ EÓ œ int] 0 ÒEÓ, that 0 Òcl\ EÓ œ cl] 0 ÒEÓ, and that 0 Ò Fr\ EÓ œ Fr] 0 ÒEÓÞ That is, 0 takes
interiors to interiors, closures to closures, and boundaries to boundaries.
Definition 8.11 A property P of topological spaces is called a topological property if, whenever
a space \ has property T and ] ¶ \ , then the space ] also has property T .
By definition: if \ and ] are homeomorphic, then \ and ] have the same topological
properties. Conversely, if two topological spaces \ and ] have the same topological properties,
then \ and ] must be homeomorphic. (Why? Let T œ “is homeomorphic to \ .” T is a
topological property because if ] has T (that is, ] ¶ \ ) and ^ ¶ ] ß then ^ also has T .
And \ has this property T , because \ ¶ \ . So if we are assuming that \ and ] have the
same topological properties, then ] has the property T , that is, ] is homeomorphic to \ .)
So we think of two homeomorphic spaces as “topologically identical”  they are homeomorphic
iff they have exactly the same topological properties. We can show that two spaces are not
homeomorphic by showing a topological property of one space that the other space doesn't
possess.
Example 8.12 Let T be the property that “every continuous real-valued function achieves a
maximum value.” Suppose a space \ property T and that 2 À \ Ä ] is a homeomorphism.
We claim that ] also has property T .
Let 0 be any continuous real-valued function defined on ] .
Then 0 ‰ 2 À \ Ä ‘ is continuous À
‘
ß Å 0
\Ò ]
2
1
By assumption, 1 œ 0 ‰ 2 achieves a maximum value at some point + − \ , and we claim
that 0 must achieve a maximum value at the point , œ 2Ð+Ñ − ] Þ If not, then there is a
point C − ] where 0 ÐCÑ  0 Ð,Ñ. Let B œ 2" ÐCÑÞ Then
1ÐBÑ œ 0 Ð2ÐBÑÑÑ œ 0 Ð2Ð2" ÐCÑÑÑÑ œ 0 ÐCÑ  0 Ð,Ñ œ 0 Ð2Ð+ÑÑ œ 1Ð+Ñ,
which contradicts the fact that 1 achieves a maximum value at +.
Therefore T is a topological property.
141
For example, the closed interval Ò!ß "Ó has property T discussed in Example 8.12 (this is a wellknown fact from elementary analysis  which we will prove later). But Ð!ß "Ñ and Ò!ß "Ñ do not
have this property T (why?). So we can conclude that Ò!ß "Ó is not homeomorphic to either Ð!ß "Ñ
or Ò!ß "ÑÞ
Some simple examples of topological properties are: cardinality, first and second countability,
Lindelof,
¨ separability, and (pseudo)metrizability are topological properties. In the case of
metrizability, for example:
If Ð\ß .Ñ is a metric space and 0 À Ð\ß .Ñ Ä Ð] ß g Ñ is a homeomorphism, then we can
define a metric . w on ] as . w Ð+ß ,Ñ œ .Ð0 " Ð+Ñß 0 " Ð,ÑÑ for +ß , − ] . You then need
to check that g. w œ g (using the properties of a homeomorphism and the definition of
. w ). This shows that Ð] ß g Ñ is metrizable. Be sure you can do this!
9. Sequences
In Chapter II we saw that sequences are a useful tool for working with pseudometric spaces. In
fact, sequences are sufficient to describe the topology in a pseudometric space Ð\ß .Ñ  because
convergent sequences determine the closure of a set.
We can easily define convergent sequences in any topological space Ð\ß g Ñ. But as we shall see,
sequences need to be used with more care in spaces that are not pseudometrizable. Whether or
not a sequence converges to a certain point B is a “local” question  it depends on “getting near
B” and we “measure nearness” to B by using the neighborhood system aB . If aB is “too large” or
“too complicated,” then it may be impossible for a sequence to “get arbitrarily close” to B. We
will see a specific example soon where such a problem occurs. But first, we look at some of the
things that do work out just as nicely for topological spaces as they do in pseudometric spaces.
Definition 9.1 Suppose ÐB8 Ñ is a sequence in Ð\ß g Ñ. We say that ÐB8 Ñ converges to B if, for
every neighborhood [ of B, b5 −  such that B8 − [ when n k. In this case we write
ÐB8 Ñ Ä B . More informally, we can say that ÐB8 Ñ Ä B if ÐB8 Ñ is eventually in every
neighborhood [ of B.
Clearly, we can replace “every neighborhood [ of B” in the definition with “every basic
neighborhood F of B” or “every open set S containing B.” Be sure you are convinced of
this.
In a pseudometric space a sequence can converge to more than one point, but we proved that in a
metric space limits of convergent sequences must be unique. A similar situation holds in any
space: the important issue is whether we can “separate points by open sets.”
142
Definition 9.2 A space Ð\ß g Ñ is a X" -space if for every pair of points B Á C − \ there exist
open sets Y and Z such that B − Y , C Â Y and C − Z ß B Â Z (that is, each of the points is an
open set that does not contain the other point).
Ð\ß g Ñ is called a X# -space (or Hausdorff space ) if whenever B Á C − \ there exist
disjoint open sets Y and Z such that B − Y and C − Z .
It is easy to check that \ is a X" -space iff for every B − \ , ÖB× is closedÞ
There are several “separation axioms” called X! ß X" ß X# ß X$ ß and X% that a
topological space might satisfy. We will look at all of them eventually. Each
condition is stronger than the earlier ones in the list Ðfor example, X# Ê X" ÑÞ
The letter “X ” is used here because in the early (German) literature, the word
for “separation axioms” was “Trennungsaxiome.”
Theorem 9.3 In a Hausdorff space Ð\ß g Ñ, a sequence can converge to at most one point.
Proof Suppose B Á C − \ . Choose disjoint open sets Y and Z with B − Y and C − Z . If
ÐB8 Ñ Ä B, then ÐB8 Ñ is eventually in Y , so ÐB8 Ñ is not eventually in Z . Therefore ÐB8 Ñ does not
also converge to C. ñ
When we try to generalize results from pseudometric spaces to topological spaces, we often get a
better insight about where the heart of a proof lies. For example, to prove that limits of sequences
are unique it is the Hausdorff property that is important, not the presence of a metric. For a
pseudometric space Ð\ß .Ñ we proved that B − cl E iff B is a limit of a sequence Ð+8 Ñ in E. That
proof (see Theorem II.5.18) used the fact that there was a countable neighborhood base ÖF " ÐBÑ À
8
8 − × at each point B. We can see now that the countable neighborhood base was the crucial
fact: we can prove the same result in any first countable topological space Ð\ß g Ñ.
First, two technical lemmas are helpful.
Lemma 9.4 Suppose ÖZ" ß Z# ß ÞÞÞß Z5 ß ÞÞÞ× is a countable neighborhood base at + − \ and define
Y5 œ int ÐZ" ∩ ÞÞÞ ∩ Z5 Ñ. Then ÖY" ß Y# ß ÞÞÞß Y5 ß ÞÞÞ × is also a neighborhood base at +.
Proof Z" ∩ ÞÞÞ ∩ Z5 is a neighborhood of +, so + − int ÐZ" ∩ ÞÞÞ ∩ Z5 Ñ œ Y5 Þ Therefore Y5 is a
open neighborhood of +. If R is any neighborhood of +, then + − Z5 © R for some 5 , so then
+ − Y5 © Z5 © R . Therefore the Y5 's are a neighborhood base at +Þ ñ
The important thing is that we can get an “improved” neighborhood base ÖY" ß Y# ß ÞÞÞß Y5 ß ÞÞÞ × in
which the Y5 's are open and Y" ª Y# ª ÞÞÞ ª Y5 ª ÞÞÞ ; the exact formula for the Y5 's in
Lemma 9.4 doesn't matter. This new neighborhood base at + plays a role like the neighborhood
base F" Ð+Ñ ª F " Ð+Ñ ª ÞÞÞ ª F " Ð+Ñ ª ÞÞÞ in a pseudometric space. We call
#
5
ÖY" ß Y# ß ÞÞÞß Y5 ß ÞÞÞ× an open, shrinking neighborhood base at +Þ
Lemma 9.5 Suppose ÖY" ß Y# ß ÞÞÞß Y5 ß ÞÞÞ× is a shrinking neighborhood base at B and that +8 − Y8
for each 8. Then Ð+8 Ñ Ä BÞ
143
Proof If [ is any neighborhood of B, then there is a 5 such that B − Y5 © [ . Since the Y5 's
are a shrinking neighborhood base, we have that for any 8 5ß +8 − Y8 © Y5 © [ . So
Ð+ 8 Ñ Ä BÞ ñ
Theorem 9.6 Suppose Ð\ß g Ñ is first countable and E © \ . Then B − cl E iff there is a
sequence Ð+8 Ñ in E such that Ð+8 Ñ Ä B.
( More informally, “sequences are sufficient” to
describe the topology in a first countable topological space.)
Proof ( É ) Suppose Ð+8 Ñ is a sequence in E and that Ð+8 Ñ Ä B. For each neighborhood [ of
B, Ð+8 Ñ is eventually in [ . Therefore so [ ∩ E Á g, so B − cl EÞ (This half of the proof works
in any topological space: it does not depend on first countability.)
( Ê ) Suppose B − cl EÞ Using Lemma 9.4, choose a countable shrinking neighborhood
base ÖY" ß ÞÞÞß Y8 ß ÞÞÞ× at BÞ Since B − cl E, we can choose a point +8 − Y8 ∩ E for each 8. By
Lemma 9.5, Ð+8 Ñ Ä B. ñ
We can use Theorem 9.6 to get an upper bound on the size of certain topological spaces,
analogous to what we did for pseudometric spaces. This result is not very important, but it
illustrates that in Theorem II.5.21 the properties that are really important are “first countability”
and “Hausdorff,” not the actual presence of a metric . .
Corollary 9.7 If H is a dense subset in a first countable Hausdorff space Ð\ß g Ñ , then
l\l Ÿ lHli! . In particular, If \ is a separable, first countable Hausdorff space, then
l\l Ÿ ii! ! œ -Þ
Proof \ is first countableß so for each B − \ we can pick a sequence Ð.8 Ñ in H such that
Ð.8 Ñ Ä B; formally, this sequence is a function 0B À  Ä H, so 0B − H Þ Since \ is Hausdorff,
a sequence cannot converge to two different points: so if B Á C − \ , then 0B Á 0C . Therefore the
function F À \ Ä H given by FÐBÑ œ 0B is one-to-one, so l\l Ÿ lH l œ lHli! Þ ñ
The conclusion in Theorem 9.6 may not be true if \ is not first countable: sequences are not
always “sufficient to determine the topology” of \  that is, convergent sequences cannot
always describe the closure of a set.
Example 9.8 (the space P)
Let P œ ÖÐ7ß 8Ñ À 7ß 8 − ™ß 7ß 8 !×, and let G4 be “the 4th column of P,” that is
G4 œ ÖÐ4ß 8Ñ − P À 8 œ !ß "ß ÞÞÞ×Þ We put a topology on P by giving a neighborhood base at each
point : À
ÖÐ7ß 8Ñ×
if : œ Ð7ß 8Ñ Á Ð!ß !Ñ
U: œ 
{F À Ð!ß !Ñ − F and C4  F is finite for all but finitely many 4× if : œ Ð!ß !Ñ
(Check that this definition satisfies the conditions in the Neighborhood Base Theorem 5.2 and
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therefore does describe a topology for P.) If : Á Ð!ß !Ñ then : is isolated in P. A basic
neighborhood of Ð!ß !Ñ is a set which contains Ð!ß !Ñ and which, we could say, contains “most of
the points from most of the columns.” With this topology, P is a Hausdorff space.
Certainly Ð!ß !Ñ − cl ÐP  ÖÐ!ß !Ñ×Ñ, but no sequence from P  ÖÐ!ß !Ñ× converges to Ð!ß !ÑÞ
To see this, consider any sequence Ð+8 ) in P  ÖÐ!ß !Ñ×:
i) if there is a column G4! that contains infinitely many of the terms +8 ß then
R œ ÐP  G4! Ñ ∪ ÖÐ!ß !Ñ× is a neighborhood of Ð!ß !Ñ and Ð+8 Ñ is not eventually
in R .
ii) if every column G4 contains only finitely many +8 's,
then R œ P  Ö+8 À 8 − × is a neighborhood of Ð!ß !Ñ and Ð+8 ) is
not eventually in R (in fact, the sequence is never in R ).
In P, sequences are not sufficient to describe the topology: convergent sequences can't show us
that Ð!ß !Ñ − cl ÐP  ÖÐ!ß !Ñ×Ñ. According to Theorem 9.6, this means that P cannot be first
countable  there is a countable neighborhood base at each point : Á Ð!ß !Ñ but not at Ð!ß !ÑÞ
The neighborhood system at Ð!ß !Ñ “measures nearness to” Ð!ß !Ñ but the ordering relationship
Ð ª Ñ among the basic neighborhoods at Ð!ß !Ñ is very complicated  much more complicated than
the neat, simple nested chain of neighborhoods Y" ª Y# ª ÞÞÞ ª Y8 ª ÞÞÞ that would form a base
at B in a first countable space. Roughly, the complexity of the neighborhood system is the reason
why the terms of a sequence can't get “arbitrarily close” to Ð!ß !ÑÞ
Sequences do suffice to describe the topology in a first countable space, so it is not surprising that
we can use sequences to decide whether a function defined on a first countable space \ is
continuous.
Theorem 9.9 Suppose Ð\ß g Ñ is first countable and 0 À Ð\ß g Ñ Ä Ð] ß g w Ñ.
continuous at + − \ iff whenever ÐB8 Ñ Ä +, then Ð0 ÐB8 ÑÑ Ä 0 Ð+ÑÞ
Then 0 is
Proof ( Ê ) If 0 is continuous at + and [ is a neighborhood of 0 Ð+Ñ, then 0 " Ò[ Ó is a
neighborhood of +. Therefore ÐB8 Ñ is eventually in 0 " Ò[ Ó, so 0 ÐÐB8 ÑÑ is eventually in [ Þ (This
half of the proof is valid in any topological space \Þ)
( É ) Let ÖY" ß ÞÞÞß Y8 ß ÞÞÞ× be a shrinking neighborhood base at +. If 0 is not
continuous at +, then there is a neighborhood [ of 0 Ð+Ñ such that for every 8, 0 ÒY8 Ó ©
Î [.
For each 8, choose a point B8 − Y8  0 " Ò[ Ó. Then (since the Y8 's are shrinking) we have
ÐB8 Ñ Ä + but Ð0 ÐB8 ÑÑ fails to converge to 0 Ð+Ñ because 0 ÐB8 Ñ is never in [ . ñ
(Compare this to the proof of Theorem II.5.22.)
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10. Subsequences
Definition 10.1 Suppose 0 À  Ä \ is a sequence in \ and that 9 À  Ä  is strictly
increasing. The composition 0 ‰ 9 À  Ä \ is called a subsequence of 0 .
0
Ò\
9 Å ß0 ‰9

If we write 0 Ð8Ñ œ B8 and 9Ð5Ñ œ 85 , then Ð0 ‰ 9ÑÐ5Ñ œ 0 Ð85 Ñ œ B85 . We write the sequence 0
informally as ÐB8 Ñ and the subsequence 0 ‰ 9 as (B85 Ñ. Since 9 is increasing, we have that
85 Ä ∞ as 5 Ä ∞Þ
For example, if 9Ð5Ñ œ 85 œ #5 , then 0 ‰ 9 is the subsequence informally written as
ÐB85 Ñ œ ÐB#5 Ñ, that is, the subsequence ÐB# ß B% ß B' ß ÞÞÞß B#5 ß ÞÞÞÑÞ But if 9Ð8Ñ œ 1 for all 8, then
0 ‰ 9 is not a subsequence: informally, ÐB" ß B" ß B" ß ÞÞÞß B" ß ÞÞÞ Ñ is not a subsequence of
ÐB" ß B# ß B$ ß ÞÞÞß B8 ß ÞÞÞ ÑÞ A sequence 0 is a subsequence of itself: just let 9Ð8Ñ œ 8.
Theorem 10.2 Suppose B − Ð\ß g Ñ. Then ÐB8 Ñ Ä B iff every subsequence ÐB85 Ñ Ä BÞ
Proof ( É ) This is clear because ÐB8 Ñ is a subsequence of itself.
( Ê ) Suppose ÐB8 Ñ Ä B and that ÐB85 Ñ is a subsequence. If [ is any neighborhood of
B, then B8 − [ whenever 8  some 8! . Since the 85 's are strictly increasing, 85  8! for all
5  some 5! Þ Therefore ÐB85 Ñ is eventually in [ , so ÐB85 Ñ Ä BÞ ñ
Definition 10.3 Suppose B − Ð\ß g Ñ Þ We say that B is a cluster point of the sequence ÐB8 Ñ if for
each neighborhood [ of B and for each 5 − , there is an 8  5 for which B8 − [ . More
informally, we say that B is a cluster point of ÐB8 Ñ if the sequence is frequently in every
neighborhood [ of B . (The underlined phrases mean the same thing.)
Definition 10.4 Suppose B − Ð\ß g Ñ and E © \Þ We say that B is a limit point of E if
R ∩ ÐE  ÖB×Ñ Á g for every neighborhood R of B  that is, every neighborhood of B contains
points of “arbitrarily close” to B but different from B.
Example 10.5.
1) Suppose \ œ Ò!ß "Ó ∪ Ö#× and E œ Ö#×Þ Then [ ∩ E Á g for every neighborhood
of #, but # is not a limit point of E since [ ∩ ÐE  Ö#×Ñ œ gÞ Each B − Ò!ß "Ó is a limit
point of Ò!ß "Ó and also a limit point of \ . Since Ö#× is open, # is not a limit point of \ .
2) In ‘, every point < is a limit point of . If E © , then E has no limit points in 
and no limit points in ‘.
3) If ÐB8 Ñ Ä Bß then B is a cluster point of ÐB8 ÑÞ More generally, if ÐB8 Ñ has a
subsequence ÐB85 Ñ Ä B, then B is a cluster point of ÐB8 ÑÞ (Why?)
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4) A sequence can have many cluster points. For example, if the sequence Ð;8 Ñ lists all
the elements of , then every < − ‘ is a cluster point of Ð;8 ÑÞ
5) The only cluster points in ‘ for the sequence ÐB8 Ñ œ ÐÐ  "Ñ8 Ñ are  " and ".
But the set ÖB8 À 8 − × œ Ö  "ß "× has no limit points in ‘Þ The set of cluster
points of a sequence is not always the same as the set of limit points of the set of terms in
the sequence! (Is one of these sets always a subset of the other? )
Theorem 10.6 Suppose Ð\ß g Ñ is first countable and that + is a cluster point of ÐB8 Ñ. Then there
is a subsequence ÐB85 Ñ Ä +.
Proof Let ÖY" ß Y# ß ÞÞÞß Y8 ß ÞÞÞ× be a countable shrinking neighborhood base at +Þ Since ÐB8 Ñ is
frequently in Y" , we can pick 8" so that B8" − Y" . Since ÐB8 Ñ is frequently in Y# , we can pick an
8#  8" so that B8# − Y# © Y" Þ Continue inductively: having chosen 8"  ÞÞÞ  8 5 so that
B85 − Y5 © ... © Y" , we can then choose 85"  85 so that B85" − Y5" © Y5 Þ Then ÐB85 Ñ is a
subsequence of ÐB8 Ñ and ÐB8 Ñ Ä +Þ ñ
Example 10.7 (the space P, revisited)
Let P be the space in Example 9.8 and let ÐB8 Ñ be a sequence which lists all the elements of
P  ÖÐ!ß !Ñ×Þ
Every basic neighborhood F of Ð!ß !Ñ is infinite, so F must contain terms B8 for arbitrarily
large 8. This means that ÐB8 Ñ is frequently in F , so Ð!ß !Ñ is a cluster point of ÐB8 ÑÞ
But no subsequence of ÐB8 Ñ can converge to Ð!ß !Ñ  because we showed earlier that no
sequence whatsoever from P  ÖÐ!ß !Ñ× can converge to Ð!ß !ÑÞ Therefore Theorem 10.6 may
not be true if the space \ is not first countable.
Consider any sequence ÐB8 Ñ Ä Ð!ß !Ñ in P. If there were infinitely many B8 Á Ð!ß !Ñ, then we
could form the subsequence that contains those terms, and that subsequence would be a sequence
in P  ÖÐ!ß !Ñ× that converges to Ð!ß !Ñ  which is impossible. Therefore we conclude that
eventually B8 œ Ð!ß !Ñ. Now let 0 À P Ä  be any bijection. If ÐB8 Ñ Ä Ð!ß !Ñ in P, then
0 ÐB8 Ñ œ 0 ÐÐ!ß !ÑÑ œ 5 eventually, so Ð0 ÐB8 ÑÑ Ä 5 œ 0 ÐÐ!ß !ÑÑ in .
The topology on  is discrete, so Ö5× is a neighborhood of 0 Ð!ß !Ñ, but 0 " ÒÖ5×Ó œ ÖÐ!ß !Ñ× is
not a neighborhood of Ð!ß !Ñ. We conclude that 0 is not continuous at Ð!ß !Ñ even though
Ð0 ÐB8 ÑÑ Ä 0 ÐÐ!ß !ÑÑ for every sequence ÐB8 Ñ Ä Ð!ß !ÑÞ
Theorem 9.9 does not apply to P: if a space is not first countable, sequences may be inadequate
to check whether a function 0 is continuous at a point.
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Exercises
E17. Suppose 0 ß 1 À Ð\ß g" Ñ Ä Ð] ß g# Ñ are continuous functions, that H is dense in \ , and
that 0 lH œ 1lHÞ Prove that is ] is Hausdorff, then 0 œ 1Þ (This generalizes the result in
Chapter 2, Theorem 5.12.)
E18.
A function 0 À Ð\ß g Ñ Ä ‘ is called lower semicontinuous if
0 " ÒÐ,ß ∞ÑÓ œ ÖB À 0 ÐBÑ  ,× is open for every , − ‘,
and 0 is called upper semicontinuous if
0 " [Ð  ∞ß ,Ñ ] œ ÖB À 0 ÐBÑ  ,× is open for each , − ‘.
a) Show that 0 is continuous iff 0 is both upper and lower semicontinuous.
b) Give an example of a lower semicontinuous 0 À ‘ Ä ‘ which is not continuous.
Do the same for upper semicontinuous.
c) Prove that the characteristic function ;E of a set E © Ð\ß g Ñ is lower semicontinuous
if E is open in \ and upper semicontinuous if E is closed in \ .
E19. Suppose \ is an infinite set with the cofinite topology, and that ] has the property that
every singleton ÖC× is a closed set. (Such a space ] is called a X" -space. Ñ Let 0 À \ Ä ] be
continuous and onto. Prove that either 0 is constant or \ is homeomorphic to ] .
1) Note: the problem does not say that if 0 is not constant, then 0 is a homeomorphism.
2) Hint: Prove first that if 0 is not constant, then l\l œ l] lÞ Then examine the topology
of ] .)
E20. Suppose \ is a countable set with the cofinite topology. State and prove a theorem that
completely answers the question: “what sequences in Ð\ß g Ñ converge to what points?”
E21. Suppose that Ð\ß g Ñ and Ð] ß g w Ñ are topological spaces. Recall that the product topology
on \ ‚ ] is the topology for which the collection of “open boxes”
U œ ÖY ‚ Z À Y − g ß Z − g w × is a base.
a) The “projection maps” 1B À \ ‚ ] Ä \ and 1C À \ ‚ ] Ä ] are defined by
1B ÐBß CÑ œ B and 1C ÐBß CÑ œ CÞ
We showed in Example 5.11 that 1B and 1C are continuous. Prove that 1B and 1C are
open maps. Give examples to show that 1B and 1C might not be closed.
b) Suppose that Ð^ß g ww Ñ is a topological space and that 0 À ^ Ä \ ‚ ] Þ Prove that
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0 is continuous iff both compositions 1B ‰ 0 À ^ Ä and 1C ‰ 0 À ^ Ä ] are
continuous.
c) Prove that ÐÐB8 ß C8 ÑÑ Ä ÐBß CÑ − \ ‚ ] iff ÐB8 Ñ Ä B in \ and ÐC8 Ñ Ä C in ] .
ÐFor this reason, the product topology is sometimes called the “topology of
coordinatewise convergence.ӄ
d) Prove that \ ‚ ] is homeomorphic to ] ‚ \Þ (Topological products are
commutative. )
e) Prove that Ð\ ‚ ] Ñ ‚ ^ is homeomorphic to
\ ‚ Ð] ‚ ^Ñ (Topological products are associative. )
E22.
Let Ð\ß g Ñ and (] ß f Ñ be topological spaces. Suppose 0 À \ Ä ] . Let
>Ð0 Ñ œ ÖÐBß CÑ − \ ‚ ] À C œ 0 ÐBÑ× = “ the graph of 0 . ”
Prove that the map 2À \ Ä >Ð0 Ñ defined by 2ÐBÑ œ ÐBß 0 ÐBÑÑ is a homeomorphism if and only if
0 is continuous.
Note: if we think of 0 as a set of ordered pairs, the “graph of 0 ” is 0 . More informally,
29A/@/<ß the problem states that the graph of a function is homeomorphic to its domain iff
the function is continuous.)
E23. In Ð\ß g Ñ, a family of sets Y œ ÖFα À α − A× is called locally finite if each point B − \
has a neighborhood R such that R ∩ Fα Á g for only finitely many α's. (Part b) was also in
Exercise E2.)
a) Suppose (\ß .Ñ is a metric space and that Y is a family of closed sets. Suppose there
is an %  ! such that . (F" , F# ) % for all F" ,F# − Y . Prove that Y is locally finite.
b) Prove that if Y is a locally finite family of sets in Ð\ß g Ñ, then
cl (α−E Fα Ñ œ α−E cl (Fα Ñ. Explain why this implies that if all the Fα 's are closed,
then α−E Fα is closed. (This would apply, for example, to the sets in part a)Þ
c) (The Pasting Lemmas: compare Exercise II.E24)
Let Ð\ß g Ñ and Ð] ß g w Ñ be topological spaces. For each α − E, suppose Fα © \ , that
0α À Fα Ä ] is a continuous function, and that 0α lÐFα ∩ F" Ñ œ 0" lÐFα ∩ F" Ñ for all αß " − E
(that is, 0α and 0" agree wherever their domains overlap). Then α−E 0α œ 0 is a function and
0 À Fα Ä ] Þ (The function 0 is built by “pasting together” all the “function pieces” 0α .)
i) Show that if all the Fα 's are open, then 0 is continuous.
ii) Give an example to show that 0 might not be continuous when there are
infinitely many Fα 's all of which are closed.
iii) Show that if there are only finitely many Fα 's and they are all closed, then 0
149
is continuous. (Hint: use a characterization of continuity in terms of closed
sets.)
iv) Show that if Y is a locally finite family of closed sets, then 0 is continuous.
(Of course, iv) Ê iii) ).
Note: the most common use of the Pasting Lemma is when the index set E is finiteÞ
For example, suppose
L" À Ò!ß "Ó ‚ Ò!ß "# Ó Ä Ð\ß .Ñ is continuousß and
L# À Ò!ß "Ó ‚ Ò "# ß "Ó Ä Ð\ß .Ñ is continuous, and
L" Ð>ß "# Ñ œ L# Ð>ß "# Ñ for all > − Ò!ß "Ó
L" is defined on the lower closed half of the box Ò!ß "Ó# ß L# is defined on the upper closed half,
and they agree on the “overlap”  that is, on the horizontal line segment Ò!ß "Ó ‚ Ö "# ×Þ Part b)
Ðor part c) Ñ says that the two functions can be pieced together into a continuous function
L À Ò!ß "Ó# Ä Ð\ß .Ñß where L œ L" ∪ L# .
150
Chapter III Review
Explain why each statement is true, or provide a counterexample. If nothing else is mentioned, \
and ] are topological spaces with no other special properties assumed.
1. For every possible topology g , the space ÐÖ!ß "ß #×ß g Ñ is pseudometrizable.
2. A convergent sequence in a first countable topological space has at most one limit.
3. If B − Ð\ß g Ñ, then ÖB× is the intersection of a family of open sets.
4. A one point set ÖB× in a pseudometric space Ð\ß .Ñ is closed.
5. Suppose g and g w are topologies on \ and that for every subset E of \ , clg ÐEÑ œ clg w ÐEÑ.
Then g œ g w Þ
6. Suppose 0 À \ Ä ] and E © \ . If 0 lE À E Ä ] is continuous, then 0 is continuous at each
point of E.
7. Suppose f is a subbase for the topology on \ and that H © \ . If W ∩ H Á g for every
W − f, then H is dense in \ .
8. If E © \ and H is dense in \ , then E ∩ H is dense in E.
9. If H is dense in Ð\ß g Ñ and g ‡ is another topology on \ with g © g ‡ , then H is dense in
Ð\ß g ‡ Ñ.
10. If 0 À \ Ä ] is both continuous and open, then 0 is also closed.
11. Every space is a continuous image of a first countable space.
12. Let \ œ Ö!ß "× with the topology g œ Ögß \ß Ö1××. There are exactly 3 continuous
functions 0 À Ð\ß g Ñ Ä Ð\ß g ÑÞ
13. If E and F are subspaces of Ð\ß g Ñ and both E and F are discrete in the subspace topology,
then E ∪ F is discrete in the subspace topology.
14. If H © ‘ and H is discrete in the subspace topology, then H is closed in ‘.
151
15. If E © Ð\ß g Ñ and \ is separable, then E is separable.
16. Suppose g is the cofinite topology on \ . Then any bijection 0 À Ð\ß g Ñ Ä Ð\ß g Ñ is a
homeomorphism.
17. If X has the cofinite topology, then the closure of every open set in X is open.
18. If Ð\ß g Ñ is metrizable, and 0 À Ð\ß g Ñ Ä Ð] ß g w Ñ is continuous and onto, then (] ß g w ) is
also metrizable.
19. If D is dense in \ , then each point of \ is a limit of a sequence of points from D.
20. A continuous bijection from ‘ to ‘ is a homeomorphism.
21. For every cardinal number 7, there is a separable topological space Ð\ß g ) with l\l œ 7.
22. If every family of disjoint open sets in Ð\ß g Ñ is countable, then Ð\ß g Ñ is separable.
23. For + − ‘# and %  !, let G% Ð+Ñ œ ÖB − ‘# À .ÐBß +Ñ œ %×.
Let g be the topology on ‘# for which the collection ÖG% Ð+ÑÀ %  !ß + − ‘# × is a subbase.
Let h be the usual topology on ‘# .
Then the function 0 À Б# ß g Ñ Ä Ð‘# ß h Ñ given by 0 ÐBß CÑ œ (sin B, sin C) is continuous.
24. If H © ‘ and each point of H is isolated in H, then cl H must be countable.
25. Consider the property W : “every minimal nonempty closed set is a singleton". Then T" Ê S.
26. A one-to-one, continuous map 0 À Ð \ß g Ñ Ä Ð] ß g Ñ is a homeomorphism iff 0 is onto.
27. An uncountable closed set in ‘ must contain an interval of positive length.
28. A countable metric space has a base consisting of clopen sets.
29. If D is dense in Ð\ß g Ñ and g ‡ is topology on \ with g ‡ © g , then D is dense in Ð\ß g ‡ Ñ.
152
30. If H © ‘ and H is discrete in the subspace topology, then H is countable.
31. The Sorgenfrey plane has a subspace homeomorphic to ‘ (with its usual topology).
32. Let U8 œ ÖF ©  À 8 − F and F © {"ß #ß ÞÞÞß 8××Þ The U8 's satisfy the conditions in the
Neighborhood Base Theorem and therefore describe a topology on .
33. At each point 8 − , let U8 œ ÖF À F ª Ö8ß 8  "ß 8  #ß ÞÞÞ ××Þ The U8 's satisfy the
conditions in the Neighborhood Base Theorem and therefore describe a topology on .
34. Suppose Ð\ß g Ñ has a base U with lU l œ -Þ Then every open cover h of \ has a subcover h
w
for which lh w l Ÿ - .
35. Suppose Ð\ß g Ñ has a base U with lU l œ -Þ Then \ has a dense set H with lHl Ÿ -Þ
36. Suppose Ð\ß g Ñ has a base U with lU l œ -Þ Then at each point B − \ , there is a
neighborhood base with lUB l Ÿ - .
37. If Ð\ß g Ñ has a base U where lU l œ 7 is finite, then lg l Ÿ #7 .
38. Suppose \ is an infinite set. Let g" be the cofinite topology on \ and let g# be the discrete
topology on \Þ If 0 À ‘ Ä Ð\ß g" Ñ is continuous, then 0 À ‘ Ä Ð\ß g# Ñ is also continuous.
39. Consider ‘ with the topology g œ ÖÐ+ß ∞Ñ À + − ‘× ∪ Ög, ‘×. Ð‘ß g Ñ is first countable.
153