Download Chapter 3 Topological and Metric Spaces

Chapter 3
Topological and Metric Spaces
The distance or more generally the notion of nearness is closely related with everyday life of any
human being so it is natural that in mathematics it plays also an important role which might be
considered in certain periods even as starring role.
Despite the historical course of affairs we start this chapter with the more general notion of
topological structure and only then we introduce the notion of metric space which turns out to be
a special case of topological structure defined on the given set. But not all topological structures
can be described by a metric because it cannot be described by convergent sequences in the considered space. A very real example of this is given by the notion of pointwise convergence of
real functions. There is no way how to introduce a metric on the space R[0,1] such that a sequence
( fn ∈ R[0,1] )n≥1 converges pointwise to a function f ∈ R[0,1] iff the distance between fn and f converges to zero.
There are many equivalent ways to develop a general theory of convergence sufficient to describe all the topological notions. We start with the notion of filter and in the second plan the
notion of net is developed. Some authors use the opposite way.
3.1 Topological spaces
Definition 3.1. Topology on a set X is a collection τ of subsets of X (i.e. τ ⊂ P(X)) having the
following properties:
(o0) ∅, X ∈ τ
(o1) A ⊂ τ ⇒ ∪A ∈ τ
(o2) A1 , . . . , An ∈ τ ⇒ ∩ni=1 Ai ∈ τ
The pair (X, τ) is called a topological space. The set X is called a carrier of the topology τ and if
no confusion occurs we will say that the set X itself is the topological space. Elements of X are
called points of the space X and the elements of τ are called open sets (in X).
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3.1 Topological spaces
41
In words, (o0) means that τ has the smallest and the largest element with respect to the set
inclusion ⊂. (o1) assures the stability of τ under arbitrary unions and (o2) means that τ is stable
under finite intersections.
Closely related concept to the open sets is that of closed sets. Given a topology τ on X a subset
B ⊂ X is called a closed set (in X relative to τ) if its complement X \ B is an open set in (X, τ). The
collection of all closed sets (relative to given topology τ on X) is denoted
τc := {c A | A ∈ τ},
or P f (X) if the topology is understood.
It is an easy exercise in de Morgan’s rules to prove the following proposition.
Proposition 3.2. The collection τc of all closed sets in the space X has the following properties:
(u0) ∅, X ∈ τc
(u1) B1 , . . . , Bn ∈ τc ⇒ ∪ni=1 Bi ∈ τc
(u2) B ⊂ τ ⇒ ∩B ∈ τc
Conversely, if F ⊂ P(X) shares the properties (u0), (u1), and (u2) then there is the unique topology
τ on X for which F coincides with the system of all subsets of X closed relative to τ.
Proof. Exercise.
So open and closed sets are the head and the tail of the same coin called topological structure
which describes the intuitive notion of nearness of elements of the set under consideration. This
nearness can be described by the more direct way using the neighbourhood notion. Let τ be a
topology (on X). If x ∈ O ∈ τ then we call the set O an open neighbourhood of the point x
and we denote by τ(x) the collection of all open neighbourhoods of the point x with respect to the
topology τ. Any set V ⊂ X which contains an open neighbourhood of x is called a neighbourhood
of x and reciprocally x is called an interior point of the set V. We denote V̊ or intV the set of all
interior points of the set V and call it interior of the set V. We denote N(x) the collection of all
neighbourhoods of the point x and call it the neighbourhood filter at x and we call (N(x) : x ∈ X)
the full neighbourhood system in X. The characteristic properties of neighbourhood systems are
given in the following proposition.
Proposition 3.3. (a) The full neighbourhood system (N(x) : x ∈ X) satisfies the following properties for all x ∈ X:
(n0) N(x) , ∅, V ∈ N(x) ⇒ x ∈ V
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3.1 Topological spaces
42
(n1) U, V ∈ N(x) ⇒ U ∩ V ∈ N(x)
(n2) N(x) 3 V ⊂ W ⊂ X ⇒ W ∈ N(x)
Any family of subsets of X verifying (n0)-(n2) will be called the filter on X.
(b) Conversely, if for each x ∈ X the collection N(x) ⊂ P(X) satisfying (n0)-(n2) is given, then
there is a unique topology τ on X for which (N(x) : x ∈ X) is the full neighbourhood system in
(X, τ).
Proof. Exercise.
Example 3.4. (a) The smallest possible topology on a set X is τ0 := {∅, X}, which is called
trivial (or indiscrete) topology on the set X.
(b) The discrete topology on X is the largest topology on X and consists of all subsets of X, i.e.
it equals to P(X).
(c) On the extended real line R̂ := R ∪ {−∞, +∞} we can define two natural topologies related
to the natural order on R̂. The lower order topology
τo := {[−∞, a) : a ∈ R̂} ∪ {R̂},
and the upper order topology
τo := {(a, +∞] : a ∈ R̂} ∪ {R̂}.
Similar order topologies can be defined on the real line R = (−∞, +∞) or more generally on
any subset of R̂, e.g. N = {1, 2, . . . }.
(d) Rather different topology on N is given by
τ := {∅} ∪ {A ⊂ N | 1 ∈ A}.
We have seen that a nonempty set can be equipped with many different topologies. The family of all topologies on X is a subset of P[P(X)] and hence is (partially) ordered by inclusion. If
σ, τ are topologies on the same set X (have the same carrier) and if τ ⊂ σ, then we say that σ is
stronger (or finer) topology that τ and we also say that τ is weaker (or coarser) topology than σ is.
So we know how to compare two topologies, but we can also construct new topological spaces
from the given topological spaces.
Let Y be a subset of the topological space X equipped with a topology τ. The trace of τ on Y is
given by
τ u Y := {O ∩ Y | O ∈ τ} =: τY ,
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3.1 Topological spaces
43
and it is easy to prove that τY is a topology on Y which is called relative topology on Y or topology
induced on Y by τ. We call (Y, τY ) a (topological) subspace of X. The sets from τY are called
relatively open sets in Y. They need not be open in X even though it may happen. If Y ∈ τ then
τY ⊂ τ but we cannot say that τY is coarser than τ because these topologies differ in their carriers.
It is obvious that relatively closed sets in Y are the traces on Y of the closed sets in X, i.e.
Y ∩ c O = Y \ (Y ∩ O).
Another operation which behaves well when applied to topological structures is the intersection of a family (τι )ι∈I of topologies on the same set X. Direct verification shows that ∩ι∈I τι =: τ
verifies the axioms (o0)-(o2) and hence (X, τ) is a topological space.
Consequently, for any family A ⊂ P(X) there exists the smallest topology on X that includes
A. This topology is denoted by τ(A) and is called topology generated by A, and A is called a
subbase of the topology τ(A). It consists of all unions of all finite intersections of the sets from
A, i.e.
nι
\
[
Vι , where Vι =
Aιk ,
O ∈ τ(A) ⇔ O =
ι∈I
k=1
with Aιk ∈ A and nι ∈ N0 := {0, 1, 2, . . . }, with the convention that the intersection of the empty
family of subsets of X equals X and its union equals the empty set ∅. This can be said more clearly
if we call a base of the topology τ any subset B ⊂ τ such that each open set O ∈ τ is the union of
some members of B, i.e. there exists B0 ⊂ B such that
[
O=
B0 .
A is a subbase of a topology τ on X iff the family


n


\




A∩ := 
A
B
⊂
X
|
∃A
,
.
.
.
,
A
∈
A
:
B
=
k
1
n




k=1
of all finite intersections of elements of A is a base of the topology τ.
There is a useful local characterization of a base B of a topology τ. Namely, B ⊂ τ is a base
for τ iff for every x ∈ X and every neighbourhood V ∈ N(x) there exists a basic open set U ∈ B
satisfying x ∈ U ⊂ V. Then we also say that
B(x) := {V ∈ B | x ∈ V}
is a neighbourhood base at the point x.
A topological space (X, τ) is called the first countable space if there exists a countable neighbourhood base at each point x ∈ X. It is called the second countable space if the topology τ has
a countable base. Obviously, the second countable space implies first countable space, but the
converse is not true in general, and you are asked to find such example in Exercises.
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3.1 Topological spaces
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Position of a point relative to a subset in topological space
In general set theory only one relationship appears between element x of a set X and a subset
A ⊂ X, either x ∈ A or x < A. In topological spaces we can make much finer classification of the
relationship between points and sets.
When x is an interior point of a set V ⊂ X, it means x ∈ V̊, we consider the points of V as the
neighbouring points of the point x and V itself is called a neighbourhood of x. A point of (X \ A)◦
is outside of A and in a sense it is far from the set A. The points of X not belonging to (X \ A)◦ are
in a sense close to A in the following sense: We say that a point x ∈ X is a closure point of a set A
in X if every neighbourhood V of x meets A, i.e. V ∩ A , ∅, and we denote Ā or clA the set of all
closure points of the set A and call it the closure of A in X. Obviously, A ⊂ Ā and Ā is the smallest
closed set containing A. Hence, Ā ⊂ A iff A is a closed subset of X. Some further properties of the
closure operation are given in Exercises.
Let A ⊂ X, then a point of X which is neither interior to A nor interior to c A is called the
boundary point of A and the set of all boundary points of A is called the boundary of A and is
denoted ∂A or Fr A (frontier of A). Obviously,
Fr A = {x ∈ X | ∀V ∈ N(x) : V ∩ A , ∅ , V ∩ Ac }.
The reader is asked to prove in Exercises the following identities
Ā = Å ∪ ∂A,
∂A = ∂(Ac ) = Ā ∩ Ac .
Closely related (but not identical) notion to the closure point is the accumulation point of a set
A ⊂ X. It is such a point x ∈ X that for each neighbourhood V of x we have
(V \ {x}) ∩ A , ∅,
i.e. any punctured neighbourhood V \ {x} meets A. We denote Ad the set of all accumulation
points of the set A and call it derived set of A.
Obviously, Ad ⊂ Ā, and the inclusion can be strict. A point x ∈ A which is not an accumulation
point of A is called the isolated point of A, and we will not use any specific notation for the set
A \ Ad of all isolated points of the set A. So, x ∈ A \ Ad iff the singleton {x} is relatively open subset
of A.
A subset S of X is called the perfect set in X if it is closed and has no isolated points, i.e.
Ā ⊂ A ⊂ Ad . Note that if A has no isolated points then its closure Ā is a perfect set. In particular, ∅
is perfect.
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3.1 Topological spaces
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Filters and nets
We have used the name neighbourhood filter for the collection N(x) of all neighbourhoods of a
point x in a topological space X. It turns out that this notion has importance in many pats of
mathematics and can be defined in any set X.
Definition 3.5. A filter on a set X is any family F of subsets of X, i.e. F ⊂ P(X), satisfying the
following properties
(f0) X ∈ F
(f1) A, B ∈ F ⇒ A ∩ B ∈ F
(f2) F 3 A ⊂ B ⊂ X ⇒ B ∈ F
A set X equipped with a filter F is called a filtered set.
If ∅ ∈ F then F = P(X) and it is called a trivial filter on X. The other filters are called nontrivial filters on X . If not specified otherwise, in the sequel by a filter on X we will understand
nontrivial filter on X, i.e. ∅ < F . Let us denote by F (X) the set of all nontrivial filters on X.
A filter F with the empty intersection, i.e. ∩F = ∅ is called a free filter. A filter which is not
free is called a fixed filter.
Example 3.6.
(a) If A , ∅ is an arbitrary fixed subset of the set X then
[A] := {M ⊂ X | A ⊂ M}
is a fixed filter on X which is nontrivial iff A , ∅. So on each nonempty set X , ∅ there are
nontrivial filters. The filter [A] is called the principal filter generated by A.
(b) Let X be an infinite set. The collection of all cofinite sets of X, i.e.
F := {M ⊂ X | M c is a finite set}
is a free filter on X.
(c) On the set N = {1, 2, . . . } the cofinite filter
F := {M ⊂ N | ∃n0 ∈ N ∀n ≥ n0 : n ∈ M}
is called Fréchet filter on N.
(d) The filter N(x) of all neighbourhoods of a given point x of a topological space.
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3.1 Topological spaces
46
The set F (X) of all nontrivial filters is partially ordered by inclusion. So two filters F , F 0 on
the same set X need not be comparable. If they are and F ⊂ F 0 , we say that filter F 0 is finer than
F . A nontrivial filter F for which there exists no (nontrivial) filter containing F as a proper subset
is called an ultrafilter on X. So ultrafilters on X are maximal elements in the partially ordered
set (F (X), ⊂). Even if the construction of ultrafilters is not an easy matter and requires the Zorn’s
lemma, we can give simple characterization of ultrafilters.
Proposition 3.7. For any filter F on X the following assertions are equivalent:
(i) F is an ultrafilter
(ii) A, B ∈ F ⇔ A ∩ B ∈ F ,
(iii) A ∈ F ⇔ Ac < F
(∀A, B ⊂ X)
(∀A ⊂ X)
Proof. Exercise.
The existence of free ultrafilters on a set X can be proved using the Zorn’s lemma only. Strange
enough, the fixed ultrafilters on any set X can be described very easily, namely the free filter [A] is
an ultrafilter iff A = {a} is singleton and then
[{a}] = {B ⊂ X | a ∈ B} =: Ua .
For the one-set family A = {A} it is easy to construct the filter containing A, it is just [A]. Not
for every family the situation is so easy. But there is a variety of families which allow to construct
easily the filter containing this family. A family of sets B ⊂ P(X) is called a filter base if
(0) ∅ < B , ∅
(1) ∀A, B ∈ B ∃C ∈ B : C ⊂ A ∩ B.
If B is a filter base then the union of all principal filters generated by elements of B, i.e. the
family
{M ⊂ X | ∃A ∈ B : A ⊂ M} =: [B]
is the smallest filter on X containing B and we call it the filter generated by B. We also say that B
is a base of the filter [B]. Any filter F itself is its base and there may exist many other bases of the
same filter F . The system τ(x) of all open neighbourhoods of x in the topology τ is the base of the
neighbourhood filter N(x) of all neighbourhoods of x.
Since the intersection of any family of filters (Fi )i∈I on the same set X is again the filter on X
(trivial if I = ∅), for every family S ⊂ P(X) there is the smallest filter on X containing S which we
call the filter generated by S and we denote it [S]ϕ . Obviously, [S]ϕ is trivial iff
∅ ∈ S∩ := {M ⊂ X | ∃A1 , . . . , Ak ∈ S : M = A1 ∩ · · · ∩ Ak }.
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3.1 Topological spaces
47
If this is not the case then S∩ is the base of the filter [S]ϕ , i.e. [S]ϕ = [Sn ] and we call S the
subbase of the generated filter [S]ϕ .
In the description of convergence issues the notion of the grill plays an important role. For any
family A of subsets of X the grill of A (relative to X) is the family
Ä := {B ⊂ X | ∀A ∈ A : A ∩ B , ∅}.
When A = [A] is a principal filter on X then
[Ä] = {B ⊂ X | A ∩ B , ∅}.
If B is a filter base then B ⊂ B̈ . The grill of the neighbourhood filter N(x) is
N̈(x) = {A ⊂ X | ∀V ∈ N(x) : V ∩ A , ∅},
which contains exactly the subsets of X which have x as the closure point, i.e.
N̈(x) = {A ⊂ X | x ∈ Ā}.
If B is the upper section filter base of a directed set (I, ), i.e.
B = {ι4 | ι ∈ I},
then
B̈ = {A ∈ I | ∀ι ∈ I ∃ι0 ∈ A : ι0 ι},
i.e. B̈ contains exactly the cofinal subsets of (I, ).
Remark 3.8. The dual notion to the filter is the notion of ideal. More precisely, a family of sets J
is called an ideal of sets if it has the following properties
(i0) ∅ ∈ J
(i1) A, B ∈ J ⇒ A ∪ B ∈ J
(i2) B ⊂ A ∈ J ⇒ B ∈ J
If J ⊂ P(Y) then we say that J is an ideal of subsets of the set Y. If ∪J = X then X is the smallest
set for which J ⊂ P(X) and we say that J is an ideal on the set X. The duality with the filters on
X is evident from the fact that J is an ideal on X iff
c
J = {Ac | A ∈ J},
is a filter on X.
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3.1 Topological spaces
48
Because of this duality both notions, filter and ideal, can be used interchangeably to describe
the convergence in a topological space. We prefer the filter approach used by Henry Cartan in 1937
in the general convergence theory. Let (X, τ) be a topological space and F be a filter on X. We say
that the filter converges to a point x ∈ X, written F → x, if N(x) ⊂ F , i.e. F is finer than the
filter of all neighbourhoods of the point x. It is equivalent to claim that τ(x) ⊂ F since the family
τ(x) of all open neighbourhoods of the point x is a base of the filter N(x). We will say that a filter
base B converges to x if the generated filter [B] → x. The notion of filter base convergence is
more convenient than the filter convergence to describe the limit notion of a function f defined on
a filtered set (X, F ) and taking values in a topological space (Y, σ). We will say that the function
f has limit b ∈ Y with respect to the filter F if the filter base
f (F ) := { f [A] | A ∈ F } → b.
We can generalize the notion of limit in two directions. First, instead of a function f we will
consider a multifunction Γ : X −−≺ Y, ι 7→ Γ(ι) =: Aι called also a filtered family and denoted
Γ = (Aι )ι∈X and second a cluster point of such filtered family will play an important role throughout
the text.
Definition 3.9. Let B be a filter base on the index set I and Γ = (Aι )ι∈I be a filtered family of
subsets in a topological space (Y, τ). A point b ∈ Y is called a cluster point of the filtered family Γ
with respect to the filterbase B if
∀V ∈ N(b) ∀B ∈ B ∃ι ∈ B : V ∩ Aι , ∅.
The set of all cluster points of Γ is called an upper topological limit of Γ and we denote it by
Ls Γ,
B
or
lim Γ.
B
Similarly, a point b ∈ Y is called a limit point of the filtered family Γ if
∀V ∈ N(x) ∃B ∈ B ∀ι ∈ B : V ∩ Aι , ∅.
The set of all limit points of Γ is called a lower topological limit of Γ and we denote it as
Li Γ,
B
or
lim Γ.
B
In the Exercises you are asked to show the following Choquet’s characterization of topological
limits




\ [ 
\ [ 
 Aι  , Ls(Aι ) =
 Aι  .
Li(Aι ) =
B
B
J∈B̈
ι∈J
K∈B
Obviously,
Li(Aι ) ⊂ Ls(Aι ),
B
48
B
ι∈K
3.1 Topological spaces
49
when the equality occurs
Li(Aι ) = Ls(Aι ) = A0 ,
B
B
we say that the filtered family of sets converges in the sense of Kuratoski-Painlevé to A0 or that A0
is the K-limit of (Aι ) and we write
A0 = K − lim(Aι ).
Example 3.10. Let X be a set with the discrete topology. Then for any sequence (An ⊂ X)n∈N we
have the following equivalences
x0 ∈ Li(An ) ⇔ ∃n0 ∈ N ∀n ≥ n0 : x0 ∈ An ,
x0 ∈ Ls(An ) ⇔ ∀n ∈ N ∃n0 ≥ n : x0 ∈ An0 ,
i.e.
Li(An ) =
[
(An ∩ An+1 ∩ . . . )
n∈N
is the Kuratowski lower limit of (An ) and
Ls(An ) =
\
(An ∪ An+1 ∪ . . . )
n∈N
is the Kuratowski upper limit of (An ).
Nets and sequences
It is well known that in metric spaces the important topological notions such as accumulation points
or closure points and continuity of functions can be described in terms of sequences convergence.
If we consider a sequence x = (x1 , x2 , . . . ) in X as the mapping
x : N → X,
and we consider Fréchet filter F = {A ⊂ N : c A is finite} on N then the sequence x converges to
x0 iff the filtered family (xn ; N, F ) converges to x0 . Unfortunately, in general topological spaces
convergent sequences do not describe all topological properties. For example x can be a closure
point of a set A in the topological space (X, τ) and no sequence of points in A will converge to x.
This inconvenience can be removed by introducing nets.
Definition 3.11. Let (I, ) be a directed set. Any mapping x : I → M is called a net in the set M
and we denote it either (xι : ι ∈ I, ) or (xι )ι∈I or simply (xι ) if the directed set (I, ) is understood
from the context. When M = P(X) a net s : I → P(X) is called a net of subsets of X.
Nets can be considered as a special type of filtered families, since on directed set (I, ) we can
consider the filter F of residual sets which is generated by the filterbase B of all upper sections
ιM := {λ ∈ I | λ ι} in I, i.e.
B := {ι4 , ι ∈ I},
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3.1 Topological spaces
50
and the generated filter is
[B] = {J ⊂ I | ∃ι ∈ I : ι4 ⊂ J} =: FI ,
and we will identify the net s with the filtered family (xι : ι ∈ I, FI ).
Using this identification we obtain the following notion of nets convergence.
Definition 3.12. (a) Let (xι ; ι ∈ I, ) =: x be a net of elements of a topological space (X, τ). A
point x0 ∈ X is called a limit point of the net x, if the associated filtered family (xι ; ι ∈ I, FI )
converges to x0 , i.e. the image filterbase x(FI ) → x0 . It means that
∀U ∈ N(x0 ) ∃ι0 ∈ I : ∀ι ι0 : xι ∈ U.
As before the set of all limit points of the net x = (xι )ι∈I is called a lower [residual] limit and will
be denoted
Li {xι },
or just Li x.
(I,)
(b) A point x0 ∈ X is called a cluster point of the net x if
∀U ∈ N(x0 ) ∀ι ∈ I ∃ι0 ι : xι0 ∈ U.
The set of all cluster points of the net x is called an upper [cofinal] limit and will be denoted
Ls {xι },
(I,)
or just
Ls x.
The main advantage of the nets over the sequences is the following characterization of the
closure.
THEOREM 3.13. Let (X, τ) be a topological space. Then for any subset A ⊂ X we have
x0 ∈ Ā ⇔ ∃(xι ∈ A : ι ∈ I, ) → x0 .
Proof. ⇐
Let us consider a convergent net
(xι ∈ A : ι ∈ I, ) → x0 .
Then for each V ∈ N(x0 ) we have ι0 ∈ I such that for all ι ι0 we have xι ∈ V ∩ A. Hence, for all
V ∈ N(x0 ) the intersection of the sets V and A is non-empty and consequently x0 ∈ Ā.
⇒ Let us consider the full neighbourhood system N(x0 ) directed by inclusion ⊂. If x0 ∈ Ā then
∀V ∈ N(x0 ) : ∃xV ∈ V ∩ A
and obviously the net
(xV : V ∈ N(x0 ), ⊂) → x0 .
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3.1 Topological spaces
51
One of the most important notions in topology is the notion of continuity of functions between
topological spaces.
Definition 3.14. Let (X, τ), (Y, σ) be topological spaces. A function f : X → Y is said to be
continuous at a point x0 ∈ X if
∀V ∈ σ[ f (x0 )] ∃U ∈ τ(x0 ) : f [U] ⊂ V,
i.e. the image filter base f (τ(x0 )) is the base of the neighbourhood filter N( f (x0 )). We say that f
is continuous on a set M ⊂ X if it is continuous at any point of this set. When M = X we simply
say that f is continuous function.
An important characterization of continuous functions is by using nets.
THEOREM 3.15. For any function f : X → Y the following statements are equivalent:
(a) f is continuous at x0 ∈ X
(b) if a net (xι ) → x0 in X then the image net ( f (xι )) → f (x0 ) in Y
(c) if a filter F → x0 in X then the image filter f (F ) → f (x0 ) in Y.
THEOREM 3.16. For any function f : X → Y the following statements are equivalent:
(i) f is continuous
(ii) if V is an open [closed] set in Y then f −1 [V] is open [closed] in X
˚
(iii) ∀B ⊂ Y : f −1 [ B̊] ⊂ f −1 [B]
(iv) ∀A ⊂ X : f [Ā] ⊂ f [A]
(v) f −1 [V] is an open subset in X for each V in a subbase of the topology on Y.
Proof. (i) ⇒ (ii) Let V be an open set in Y and consider any x ∈ f −1 [V]. Then from the continuity
of f at x there is a neighbourhood U ∈ N(x) such that
f [U] ⊂ V,
i.e.
U ⊂ f −1 [V],
which gives that x is an interior point of f −1 [V]. Hence f −1 [V] is open being the neighbourhood
of all its point. If V is closed in Y then f −1 [Y \ V] = X \ f −1 [V] is open in X and therefore f −1 [V]
is closed in X.
(ii) ⇒ (iii) Since B̊ ⊂ B is open, f −1 [ B̊] is an open subset of f −1 [B] and the claim follows.
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3.2 Metric spaces
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(iii) ⇒ (iv) Let y = f (x) ∈ f [Ā] for some x ∈ Ā. Then for each open neighbourhood of y we have
˚
f −1 [V] = f −1 [V̊] ⊂ f −1 [V], so f −1 [V] is an open neighbourhood of x. Therefore, f −1 [V] ∩ A , ∅,
so V ∩ f [A] , ∅ and y ∈ f [A] follows.
(iv) ⇒ (v) Let V be an open subset of Y. Put A := ( f −1 [V])c = f −1 [V c ]. Then we have
f [Ā] = f [A] = f [ f −1 [V c ]] ⊂ V c = V c ,
so Ā ⊂ f −1 [V c ] = A. And consequently, A = Ā, so that A is closed set and hence f −1 [V] = Ac is
open.
(v) ⇒ (i) Since the topology on Y is its subbase the preimage of any open set in Y is an open set in
X which implies immediately that f is continuous at any point of its domain X.
3.2 Metric spaces
A very convenient way how to introduce a topology on a set X is by using metric on X which is a
nonnegative real function d : X × X → R+0 that measures the distance between any pair of points
of X and satisfies the following properties (for all x, y, z ∈ X):
(m0) d(x, y) = 0 ⇒ x = y
(m0’) x = y ⇒ d(x, y) = 0
(m1) d(x, y) = d(y, x)
(m2) d(x, y) ≤ d(x, z) + d(z, y)
The pair (X, d) is called a metric space. In a metric space (X, d) we can introduce the topology
associated with the metric d. For any r > 0 and x ∈ X the (open) ball centred at x with radius r is
defined by
B(x, r) := {y ∈ X : d(x, y) < r}.
The system
{B(x, r) : r ∈ Q+ }
is the countable neighbourhood base at x and the corresponding topology is called a metric topology associated with the metric d and will be denoted by τd .
A topology on a set X can be introduced by more general function than a metric. An extended
metric on X is a function d : X × X → [0, +∞] which satisfies all the axioms (m0)-m(2) for a
metric with the convention ∞ + a = +∞ for any a ∈ [0, +∞]. An [extended] pseudometric on X is
a (extended) real function d : X ×X → [0, +∞]which satisfies the axioms for (extended) metric except (m0). A(n) (extended) hemimetric on X is a(n) (extended) real function d : X × X → [0, +∞]
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3.2 Metric spaces
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which shares only the properties (m0’) and (m2), i.e hemimetric need not be a symmetric function.
For (extended) hemimetric [pseudometric, metric] d we can still define the ball centered at x
and with radius r
B(x, r) := {y | d(x, y) < r}
keeping the convention of the order in the definition of a ball with respect to a hemimetric which
is not symmetric.
The proof of the next proposition is left as an exercise.
Proposition 3.17. Let (X, d) be a hemimetric space. If we define an interior point x ∈ V as such
a point for which ∃B(x, r) ⊂ V and declare V open if all its points are interior points of the set V
then the set of all open sets is a topology on X and an open ball is an open set.
Let (X, d) be a metric space. A subset A ⊂ X can be expanded by a factor λ > 0 if we define
[
B(a, λ) = {x ∈ X | d(x, A) < λ}.
λ + A :=
a∈A
If A ⊂ λ + B for some λ > 0 then we say that A is not much larger than B. Then the well-defined
function
hl (A, B) := inf{λ ≥ 0 | A ⊂ B + λ}
is hemimetric on P(X) and the associated topology on P(X) is called the lower hemimetric topology.
Similarly, the phrase A is not much smaller than B can be translated as
∃ > 0 : B ⊂ + A.
Again we can construct a hemimetric
hu (A, B) := hl (B, A),
and the associated topology is called the upper hemimetric topology.
The Hausdorff extended pseudometric on P(X) is defined by
h(A, B) := max{hl (A, B), hu (A, B)},
and the topology generated by this pseudometric is called the Vietoris topology.
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