Download Examples of topological spaces

Examples of topological spaces
John Terilla
Fall 2014
Contents
1
Introduction
1
2
Some simple topologies
2
3
Metric Spaces
2
4
A few other topologies on R1 and R2 .
4
5
The Zariski Topology
4
6
Continuous functions
4
7
Limit and boundary points
6
8
Sequences and separation
6
9
Sequences and closure
8
10 Sequences and continuity
8
11 Separation, closure, and continuity
9
12 Sequences and first countable spaces
9
1
Introduction
I’m assuming that students know linear algebra and have had some abstract algebra,
for example, principal ideals appear in these notes without definition. Later, I’ll be
assuming students know what a group is, how to define the quotient of a group by a
normal subgroup, etc...
1
I’m also assuming that students have some basic working knowledge about sets.
Notions such as function and countable appear in these notes without definition. It’s
assumed that students know these things, as well as concepts such as equivalence relation, partial order, Cartesian product, surjective, etc ... There’s usually a review of
basic set theory in the first chapter of a general topology book, for example Chapter 0
of [2] or Chapters 1, 2, 3 in [3].
2
Some simple topologies
Example 1. Let X be any set. The discrete topology on X is defined to be 2X . The
indiscrete topology on X is defined to be {∅, X}.
Problem 1. Draw a diagram of all the topologies on a three point set indicating which
are contained which.
Problem 2. The integers Z are given the discrete topology unless specified otherwise.
There’s another topology on Z for which sets
aZ + b = {a + nb : n ∈ N}
for a ∈ Z\{0} and b ∈ Z, together with ∅, are open. Furstenberg [1] used this topology
in a delightful proof that there are infinitely many primes. One can check that the sets
aZ + b are also closed in this topology. Since every integer except ±1 has a prime
factor
[
Z \ {−1, 1} =
S(p, 0).
primes p
Since the left hand side is not closed (no nonempty finite set can be open) there must
be infinitely many closed sets in the union on the right. Therefore, there are infinitely
many primes.
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Metric Spaces
A metric space is a pair (X, d) where X is a set and d : X × X → R satisfies
• d(x, y) ≥ 0 for all x, y ∈ X,
• d(x, y) = d(y, x) for all x, y ∈ X,
• d(x, y) + d(y, z) ≤ d(x, z) for all x, y, z ∈ X
• d(x, y) = 0 if and only if x = y for all x, y ∈ X.
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The function d is called a metric. If (X, d) is a metric space, x ∈ X, and r > 0, the
ball centered at x of radius r is defined to be
B(x, r) = {y ∈ X : d(x, y) < r} .
The balls {B(x, r)} form a basis for a topology on X called the metric topology.
Example 2. For any x, y ∈ Rn ,
v
u n
uX
d(x, y) = t (xi − yi )2
i=1
where x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) defines the usual metric on Rn and the
induced metric topology on Rn is called the usual topology on Rn .
Example 3. Let C([0, 1]) denote the set of continuous functions on [0, 1]. The following define metrics on C([0, 1]):
d(f, g) = sup |f (x) − g(x)|.
x∈[0,1]
Z
1
|f − g|.
ρ(f, g) =
0
Example 4. Let l2 be the set of all the real valued sequences {xn } for which
converges. Then d : l2 × l2 → R defined by
v
u∞
uX
(xn − yn )2
d ({xn }, {yn }) = t
P∞
n=1
x2n
n=1
defines a metric.
p
Example 5. In any normed vector space, the function d(x, y) = kx − yk defines a
metric. Some of the examples above are metrics arising this way. But there are metric
spaces that are not linear. For example, if (X, d) is a metric space and Y is any subset
of X, (Y, d) will be a metric space, where d is the metric on X restricted to Y × Y .
Problem 3. Let (X, d) be a metric space. Show that for any x ∈ X and any r > 0, the
set {y ∈ X : d(x, y) ≤ r} is closed. Give an example to show that
{y ∈ X : d(x, y) ≤ r} may not equal B(x, r).
Problem 4. Prove that every metric space is first countable. See definition 6 in these
notes.
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4
A few other topologies on R1 and R2 .
Example 6. Check that the sets [a, b) for a < b form a basis for a topology on R.
This topology is called the lower limit topology, or the Sorgenfrey topology, or the
uphill topology, or the half-open topology, and it probably goes by other names too.
Compare this lower limit topology to the ordinary topology.
Example 7. In general, the intervals (a, b) = {x ∈ X : a < x < b} define a
topology on any totally ordered set (include the intervals (a, ∞) and (−∞, b)). The
set R is totally ordered and the order topology on R coincides with the usual topology.
The lexicographic order makes R2 totally ordered. Compare the lexicographic order
topology on R2 to the ordinary one.
Example 8. Any set X has a cofinite topology where a set U is open if and only X \ U
is finite (or if U = ∅). The open sets in the cocountable topology are those whose
complement is countable. Compare the cofinite and cocountable topologies on R to
the usual one.
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The Zariski Topology
Problem 5. Let R be a ring (commutative, with 1) and let spec(R) denote the set of
prime ideals of R. The Zariski topology on spec(R) is defining the sets
V (E) = {p ∈ spec(R) : E ⊂ p}
for any E ⊂ R to be closed. Check that these closed sets do define a topology on
spec(R) and sketch a picture of spec(C[x]) and spec(Z).
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Continuous functions
Definition 1. A function f : X → Y between metric spaces is called continuous at a
point x ∈ X if and only if for every > 0 there exists a δ > 0 so that d(f (x), f (x0 )) <
whenever d(x0 , x) < δ. Equivalently, f is continuous at x if and only if for every
> 0 there exists a δ > 0 so that B(x, δ) ⊆ B(f (x), ). The function f is called
continuous if it is continuous at every point x ∈ X.
The concept that f : X → Y is continuous can be defined without reference to the
points of X. Here’s a slightly better definition.
Better Definition. A function f : X → Y is continuous if and only if for every open
set U ⊂ Y , f −1 (U ) is open in X.
It’s left as an exercise to check that both definitions are equivalent. Although they
are equivalent, the second one is better because it makes sense for any function f :
X → Y between topological spaces. It’s redundant, but for the record:
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Definition 2. A function f : X → Y between two topological spaces is continuous if
and only if for every open set U ⊂ Y , f −1 (U ) is open in X.
The argument that shows the equivalence between the first definition and the better
definition of continuous functions between metric spaces generalizes and proves the
following:
Proposition 1. A function f : X → Y between topological spaces is continuous if
and only if f −1 (B) is open for every B in a basis for the topology on Y .
Here are a few examples
Example 9. Let X be a topological space. If S is any set with the discrete topology,
then any function f : S → X is continuous. For example, a sequence x : N → X
is continuous. (Unless otherwise specified, the natural numbers N is given the discrete
topology = metric topology on N inherited from the metric topology on R). As in other
subjects, if x : N → X is a sequence, the notation xn is used to denote x(n) and {xn }
may be used to denote the sequence x : N → R.
Example 10. If (X, d) is a metric space and x ∈ X, then the function f : X → R
defined by f (y) = d(x, y) is continuous.
One way to study a topological space X is to study the continuous functions from
X or the continuous functions to X. For example, the fundamental group of X looks
at functions from the circle to X. Also, the set hom(X, R) of continuous functions
from X to R is often interesting. Using the definition of addition and multiplication
in R, the set hom(X, R) becomes an algebra. It’s a good exercise to check: (1) that
constant functions are continuous, (2) the sum and product of continuous functions is
continuous.
Theorem 1. For any topological space X, the idenity idX : X → X is continuous.
For any topological spaces X, Y, Z and any continuous functions f : X → Y and
g : Y → Z, the composition g ◦ f : X → Z is continuous.
This theorem, though easy to prove, is important since it establishes that topological
spaces and continuous functions form a category. The set of continuous functions from
X to Y is denoted hom(X, Y ). Since topological spaces forms a category, there is a
definition of the equivalence.
Definition 3. A continuous function f : X → Y is an equivalence, or homeomorphism, if there exists a continuous function g : Y → X satisfying f ◦ g = idY and
g ◦ f = idX . In this case, the spaces X and Y are called equivalent, or homeomorphic.
Topology, then, is essentially the study of properties that are preserved under homeomorphisms. Such properties are called topological.
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Example 11. Not every continuous bijection is a homeormorphism. For example, the
identity function id : (R, τdiscrete ) → (R, τusual ) is a continuous bijection that is not
a homeomorphism.
Consistent with the themes of category theory, a topological space X is determined
by the continuous functions from X. To see this, let S = {0, 1} with the topology
{∅, {1}, S}—S is called the Serpinski space. Now, for any open set U ⊂ X, the
function χU : X → S defined by
(
1 if x ∈ U,
χU (x) =
0 if x ∈
/U
is a continuous function and every continuous function from X → S is a χU for some
open set U . Therefore, a “copy” of the entire topology of X is encoded in the set
hom(X, S).
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Limit and boundary points
For easy reference, a few standard definitions are stated here.
Let A be a subset of a topological space X. The closure of A is the smallest closed
set containing A and we denote it by A. The interior of A is the largest open set
contained in A and we denote it by A◦ . A point x ∈ X is a boundary point of A iff
every open set containing x contains both a point of A and a point of X \ A. We denote
the set of boundary points of A by ∂A. A point x ∈ X is a limit point of A iff every
open set containing x contains a point of A, not equal to x. We denote the set of limit
points of A by A0 .
◦
A is called dense if A = X and is called nowhere dense if A = ∅.
Problem 6. Here’s a clever problem from [2], due originally to Kuratowski. Let A
be a subset of a topological space X. What is the maximum number of distinct sets
that can be obtained by iteratively applying complement and closure? Find a subset A
of the real numbers that realizes this maximum number. Note: The number is finite,
although that is not obvious.
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Sequences and separation
Recall
Definition 4. Let X be a topological space. A sequence in X is a function x : N → X.
We usually write xn for x(n) and may denote the sequence {xn }. A sequence {xn }
converges to z ∈ X if and only if for every open set U containing z, there exists an
N ∈ N so that if n ≥ N , xn ∈ U. If {xn } converges to z ∈ X we write {xn } → z.
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Definition 5. Three fundamental separation axioms:
• A topological space X is T0 iff for every pair of points x, y ∈ X there exists an
open set containing one, but not both of them.
• A topological space X is T1 iff for every pair of points x, y ∈ X there exists
open sets U and V with x ∈ U , y ∈ V with x ∈
/ U and y ∈
/ V.
• A topological space X is T2 , or Hausdorff, iff for every pair of points x, y ∈ X
there exists open sets U and V with x ∈ U , y ∈ V with U ∩ V = ∅.
Observe that each separation axiom defines a topological property. Now, a few
examples:
Example 12. Let A = {1, 2, 3} with the topology τ = {∅, {1}, {1, 2}, A}. Then the
constant sequence 1, 1, 1, 1, . . . converges to 1; it also converges to 2 and to 3. Observe
that A is not T1 : there is no open set around 1 separating it from 2.
Example 13. Consider Z with the cofinite topology. For any m ∈ Z, the constant
sequence m, m, m, . . . converges to m and only to m. For if l 6= m, the set R \ m is
an open set around l contiaining no elements of the sequence.
The sequence {n} = 1, 2, 3, 4, . . . converges to m for every m ∈ Z. To see this,
let m be any integer and let U be a neighborhood of m. Since Z \ U is finite, there
can only be finitely many natural numbers in R \ U . Let N be larger than the greatest
natural number in R \ U . Then, if n > N , n ∈ U , proving that {n} → m.
Example 14. Consider R with the usual topology. If {xn } → x, then {xn } does not
converge to any number y 6= x. To prove it, we can find disjoint open sets U and V
with x ∈ U and y ∈ V (we can be explicit if necessary: U = (x − c, x + c) and
V = (y − c, y + c) where c = 12 |x − y|). Then, there is a number N so that xn ∈ U
for all n ≥ N . Since U ∩ V = ∅, V cannot contain any xn for n ≥ N and hence {xn }
does not converge to y.
Now, a few theorems:
Theorem 2. If X is T1 , then for any x ∈ X, the constant sequence x, x, x, . . . converges to x and only to x.
Proof. Suppose X is T1 and x ∈ X. It’s clear that x, x, x, . . . → x. Let y 6= x.
Then there exists an open set U with y ∈ U and x ∈
/ U . Therefore, x, x, x, . . . cannot
converge to y.
Theorem 3. In a Hausdorff space, limits of sequences are unique.
Proof. Let X be Hausdorff, let {xn } be a sequence with {xn } → x and {xn } → y.
If x 6= y, then there are disjoint open sets U and V with x ∈ U and y ∈ V. Since
{xn } → x there is a number N so that xn ∈ U for all n ≥ N . Since {xn } → y there
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is a number K so that xn ∈ U for all n ≥ K. Let M = max N, K. Since M ≥ N
and M ≥ K we have xM ∈ U and xM ∈ V contradicting the fact that U and V are
disjoint.
Now, in fact, T1 spaces are characterized by the property that constant sequences
have only one limit.
Theorem 4. If X is any topological space for which every constant sequence x, x, x, . . .
converges to x and only to x, then X is T1 .
Proof. If X is not T1 , there exist two distinct points x and y for which every open set
around y contains x. Then the sequence x, x, x, . . . → y.
One might speculate that Hausdorff spaces are characterized by the property that
convergent sequences have unique limits, but this isn’t true. See Theorem 10.
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Sequences and closure
Let X be a topological space and A ⊂ X.
Theorem 5. If {xn } is a sequence in A that converges to x, then x ∈ A.
Proof. Suppose {xn } → x and each xn ∈ A. Recall that A = A ∪ A0 . So it suffice
to prove that if x ∈
/ A, then x ∈ A0 . Let U be a neighborhood of x. Since {xn } → x,
there exist infinitely many, and in particular one, xn ∈ U . This shows that there exists
an element of A, not equal to x, in every open set around x. That is, x ∈ A0 .
The converse of Theorem 5 is false.
Example 15. Consider R with the cocountable topology and let A = [0, 1]. Then
A = R but there’s no sequence in A that convergest to 10, for example.
Theorem 6. A closed set contains the limits of all its convergent sequences
Proof. This follows from Theorem 5 since a closed set equals its closure.
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Sequences and continuity
Let X and Y be topological spaces and f : X → Y be a function.
Theorem 7. If f is continuous and {xn } is a sequence in X with {xn } → x, then
{f (xn )} → f (x).
Proof. Suppose f is continuous and {xn } → x. Let U be an open set containing
f (x). Since f is continuous, there exists an set V containing x with f (V ) ⊂ U .
Since {xn } → x, there exists a natural number N so that n ≥ n ⇒ xn ∈ V . Then
f (xn ) ∈ U for n ≥ N , proving that {f (xn )} → f (x).
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Separation, closure, and continuity
Here’s a good problem. It doesn’t use sequences, but it does nicely involve separation,
closure, continuity.
Problem 7. Let X and Y be topological spaces and suppose f : X → Y is continuous,
surjective, and open. Prove that Y is Hausdorff if and only if
{(x, x0 ) ∈ X × X : f (x) = f (x0 )}
is closed in X × X.
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Sequences and first countable spaces
Sequences are important tools in analysis. Indeed much of analysis can be characterized using sequences. For example,
Theorem 8. f : Rn → Rm is continuous if and only if for every sequence {xn } in Rn
with {xn } → x, the sequence {f (xn )} → f (x).
A common way to show that a given function is not continuous is to find a sequence
{xn } converging to a point x ∈ Rn for which the sequence {f (xn )} doesn’t converge
to f (x).
Comparing Theorems 7 and 8, one sees that for the topological spaces X = Rn
and Y = Rm , the converse to Theorem 7 is true, but the converse of Theorem 7 is not
true in general. There are spaces X and Y and discontinuous functions f : X → Y
for which {f (xn )} → f (x) for every convergent sequence {xn } → x. Before getting
into examples, let us first isolate the important topological property possessed by Rn
that makes the converse of Theorem 7 true. The feature is that Rn has a countable
neighborhood base:
Definition 6. We say a topological space X is first countable if for every x ∈ X, there
exists a countable collection {Ux,n }∞
n=1 of neighborhoods of x with the property that if
U is any neighborhood of x, there exists a Ux,n with Ux,n ⊂ U. The collection {Ux,n }
is called a countable neighborhood base.
Also, see definition 3.4 in [4]. Observe that being first countable is a topological
property.
Example 16. Every metric space is first countable. To see this, let X be a metric space
and recall that a set U is open in the metric topology if and only if for every x ∈ U ,
there exists an > 0 so that B(x, ) ⊂ U . Since for any > 0, there exists a natural
number n with 0 < n1 < , we see that the collection
1
Ux,i = B x,
, for x ∈ X and n ∈ N
n
defines a countable neighborhood base for X.
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Theorem 9. Suppose X and Y are first countable. Then f : X → Y is continuous if
and only if for every sequence {xn } in X with {xn } → x, the sequence {f (xn )} →
f (x).
Proof. It was already proved that if f is continuous and {xn } is a sequence in X with
{xn } → x then {f (xn )} → f (x). The only thing to prove is the converse.
So, suppose f is not continuous. So, there is a point x ∈ X and a neighborhood U
of f (x) so that for every neighborhood V of x, f (V ) * U. Let {Vx,i } be a neighborhood base of x. For each n ∈ N, we have f (Vx,1 ∩ · · · ∩ Vx,n ) * U , so there exists
a point xn ∈ Vx,1 ∩ · · · ∩ Vx,n with f (xn ) ∈
/ U. By this construction {xn } → x, and
{f (xn )} 9 f (x).
Since Rn is a metric space, it’s first countable so Theorem 8 is simply a special case
of Theorem 9. One might summarize the above discussion and the previous theorem as
saying that sequences suffice to describe continuous functions between first countable
spaces. More is true: sequences essentially characterize the topology of first countable
spaces.
Theorem 10. Let X be a first countable space. Then X is Hausdorff if and only if the
limits of convergent sequences are unique.
Proof. It was already proved that if X is Hausdorff, the limits of convergent sequences
are unique. So, suppose that X is not Hausdorff. Then, there exist two points x, y ∈ X
for which every neighborhood of x intersects every neighborhood of y. Let {Ux,i } and
{Vx,i } be countable neighborhood bases of x and y guaranteed by the first countability
of X. We construct a sequence {zn } as follows: for each n ∈ N, Ux,1 ∩ · · · ∩ Ux,n is
a neighborhood of x and Vx,1 ∩ · · · ∩ Vx,n is a neighborhood of y and therefore they
must intersect. Choose a point zn ∈ Ux,1 ∩ · · · ∩ Ux,n ∩ Vx,1 ∩ · · · ∩ Vx,n .
Exercise: verify that the sequence {zn } converges to both x and y.
Theorem 11. Let X be a first countable space and let A ⊂ X. A point x ∈ A if and
only if there exists a sequence {xn } in A with {xn } → x.
Proof. It was already proved that if {xn } is a sequence in A with {xn } → x, then
x ∈ A. So suppose that x ∈ A. If x ∈ A, then the constant sequence x, x, x, . . . is a
sequence in A converging to x. If x ∈ A0 then we find a sequence {xn } converging to x
using a countable neighborhood base {Ux,i }. For each n ∈ N, the set Ux,1 ∩ · · · ∩ Ux,n
is an open set containing x. Because x ∈ A0 , there exists a point of A, call it xn , with
xn ∈ Ux,1 ∩ · · · ∩ Ux,n . The remaining check that the sequence thus defined converges
to x is left as an exercise.
Theorem 12. A set in a first countable space is closed if and only if it contains the
limits of all its convergent sequences.
Proof. This is an immediate corollary of the previous theorem.
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References
[1] H. Furstenberg. On the infinitude of primes. Amer. Math. Monthly, 62(5):353, May
1955.
[2] J. L. Kelley. General Topology. Van Nostrand, 1955.
[3] Seymour Lipschutz. Schaum’s Outline of Theory and Problems of General Topology. McGraw-Hill, 1965.
[4] J.P. May.
An outline summary of basic point
http://www.math.uchicago.edu/ may/MISC/Topology.pdf.
[5] S. Willard. General Topology. Addison-Wesley, 1968.
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set
topology.