MA2223: TOPOLOGICAL PROPERTIES OF METRIC
SPACES
Contents
1. Topological spaces
1.1. Examples of topological spaces
1.2. Hausdorff topological spaces
1.3. The subspace and product topologies
1.4. Connected topological spaces
1.5. Compact topological spaces
1.6. Uniformly continuous mappings
1.7. Sequentially compact and totally bounded spaces
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MA2223: TOPOLOGICAL PROPERTIES OF METRIC SPACES
1. Topological spaces
In our study of metric spaces we made use of the notion of open sets. We
called the collection of all open sets in a metric space the metric topology.
It can often happen that two different metrics generate the same metric
topology. We saw this when studying equivalent norms. In this section we
take a closer look at properties which depend on the underlying collection
of open sets. To do this we introduce the concept of a topological space.
Definition 1.1. A topological space is a pair (X, τ ) consisting of a set X
and a collection τ of subsets of X such that
(i) ∅ and X are in τ ,
(ii) if {Uα } is a collection of sets in τ then the union
S
α
Uα is in τ ,
(iii) if U1 , . . . , Un is a finite collection of sets in τ then the intersection
Tn
j=1 Uj is in τ .
The collection τ is called a topology on X. The elements of τ are called
open sets.
Much of the terminology introduced for metric spaces can be generalised
to topological spaces (X, τ ).
A subset A of X is called a closed set if its complement X\A is an open
set.
Given two topological spaces (X, τ ) and (Y, τ 0 ), a mapping T : X → Y is
called continuous if for each open set U in (Y, τ 0 ) the preimage T −1 (U ) is
open in (X, τ ).
Motivation for these definitions comes from [MA2223 Metric spaces: Theorems 1.24, 1.31 and 1.37].
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1.1. Examples of topological spaces.
Example 1.2. The metric topology.
Let (X, d) be a metric space and let τ be the metric topology.
Example 1.3. The usual topology on Rn .
The metric topology on Euclidean space Rn is known as the usual topology
on Rn .
Example 1.4. The trivial topology.
Let X be a non-empty set and let τ = {∅, X}.
Example 1.5. The discrete topology.
Let X be a non-empty set and let τ be the collection of all subsets of X.
Example 1.6. The Sierpinski space.
Let X = {a, b} and let τ = {∅, {a}, X}.
Example 1.7. The Zariski topology.
Let X be a non-empty set and let τ be the collection of all subsets U such
that X\U is a finite set of points together with the empty set ∅. This
topology has applications in algebraic geometry.
Our primary examples of topological spaces will be metric spaces. However there are also interesting examples which are not metric spaces. A
topological space (X, τ ) is metrizable if there exists a metric d on X which
generates the topology τ . If X contains at least two elements then the
trivial topology on X is not metrizable. The Sierpinski space and Zariski
topology are not metrizable. The discrete topology on a set X is metrizable
since it is generated by the discrete metric on X.
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MA2223: TOPOLOGICAL PROPERTIES OF METRIC SPACES
1.2. Hausdorff topological spaces.
Definition 1.8. A topological space (X, τ ) is called Hausdorff if for every
pair of distinct points x, y ∈ X there exist disjoint open sets U and V with
x∈U
and y ∈ V
Example 1.9. Every metric space (X, d) is a Hausdorff topological space
(with respect to the metric topology).
Definition 1.10. Let (X, τ ) be a topological space. An open set U which
contains a point x ∈ X is called a neighbourhood of x. A sequence (xn )∞
n=1 of
points in X is said to converge to a point x ∈ X if given any neighbourhood
U of x, there exists N ∈ N such that
xn ∈ U
for all n ≥ N
We call x a limit of the sequence.
This generalizes our earlier definition of a convergent sequence in a metric
space [MA2223 Metric spaces: Definition 1.27]. We have simply replaced
open balls (which require a metric) with neighbourhoods.
Theorem 1.11. Let (X, τ ) be a Hausdorff topological space. Then
(i) subsets of X which contain a single point are closed,
(ii) every sequence in X converges to at most one limit.
Example 1.12. Consider R with the Zariski topology. The sequence (n)∞
n=1
converges to every point in R and so the Zariski topology is not Hausdorff.
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1.3. The subspace and product topologies.
Definition 1.13. Let (X, τ ) be a topological space and let A be a subset
of X. Then the collection
τA = {A ∩ U : U ∈ τ }
is a topology on A. The topology τA is called the subspace topology on A
and we call (A, τA ) a subspace of (X, τ ).
Theorem 1.14. Let (X, τ ) be a topological space and let (A, τA ) be a subspace of (X, τ ).
(i) The inclusion mapping i : A → X, x 7→ x is continuous.
(ii) If (Y, τ 0 ) is a topological space and f : X → Y is a continuous mapping
then the restriction f |A : A → Y , x 7→ f (x) is continuous.
Definition 1.15. Let (X, τ ) and (Y, τ 0 ) be topological spaces. Let β be
the collection of subsets of X × Y of the form U × V where U is an open
set in (X, τ ) and V is an open set in (Y, τ 0 ). The topology generated by β
on X × Y is called the product topology. The product topology consists of
subsets of X × Y which are unions of elements of β.
Theorem 1.16. The product topology on Rn (where we take R with the
usual topology) is the same as the usual topology on Rn .
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MA2223: TOPOLOGICAL PROPERTIES OF METRIC SPACES
1.4. Connected topological spaces.
Definition 1.17. A topological space (X, τ ) is said to be connected if there
does not exist a pair U, V of non-empty disjoint open sets with X = U ∪ V .
Theorem 1.18. Let (X, τ ) be topological space. The following statements
are equivalent,
(i) (X, τ ) is connected,
(ii) the only subsets of X which are both open and closed are the empty set
∅ and X,
(iii) every continuous function f : X → {0, 1} is constant.
Example 1.19. X = [0, 1] ∪ [2, 3] is not a connected subspace of R since
we can take U = [0, 1] and V = [2, 3].
Theorem 1.20. Every closed interval [a, b] in R is connected.
Theorem 1.21. Let (X, τ ) and (Y, τ 0 ) be topological spaces and T : X → Y
a continuous mapping. If (X, τ ) is connected then T (X) is a connected
subspace of (Y, τ 0 ).
Theorem 1.22. If (X, τ ) and (Y, τ 0 ) are connected topological spaces then
X × Y is connected (with respect to the product topology).
Theorem 1.23. If A is a connected subspace of a metric space (X, d) then
the closure of A is also a connected subspace of (X, d).
Definition 1.24. A topological space (X, τ ) is called path-connected if for
every pair of points x, y ∈ X there exists a continuous map α : [0, 1] → X
with α(0) = x and α(1) = y. The map α is called a path from x to y.
Theorem 1.25. Every path-connected topological space is connected.
Example 1.26. Normed vector spaces are path-connected. If x, y ∈ X
then we can construct the “line segment” joining x to y,
α : [0, 1] → X,
t 7→ (1 − t) x + t y
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Note that α is continuous since it is composed of addition and scalar multiplication, both of which are continuous in a normed vector space.
A connected component of a topological space (X, τ ) is a connected subspace which is maximal in the sense that it is not contained in any larger
connected subspace. Every topological space is a disjoint union of its connected components.
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MA2223: TOPOLOGICAL PROPERTIES OF METRIC SPACES
1.5. Compact topological spaces.
Definition 1.27. Let (X, τ ) be a topological space. A collection {Uα } of
open sets is called an open cover of X if
X=
[
Uα
α
A topological space (X, τ ) is called compact if every open cover {Uα } of X
contains a finite subcollection {U1 , . . . , Un } with
X=
n
[
Uj
j=1
Lemma 1.28. Let (A, τA ) be a subspace of a topological space (X, τ ). Then
(A, τA ) is compact if and only if for every collection {Uα } of open sets in
(X, τ ) with
A⊆
[
Uα
α
there exists a finite subcollection {U1 , . . . , Un } with
A⊆
n
[
Uj
j=1
Theorem 1.29. (Heine-Borel) Every closed interval [a, b] in R is compact.
Theorem 1.30. Let (X, τ ) and (Y, τ 0 ) be topological spaces and T : X → Y
a continuous mapping. If A is a compact subspace of X then T (A) is a
compact subspace of Y .
Theorem 1.31. Let (X, τ ) be a Hausdorff topological space. If A is a
compact subspace of X then A is closed.
Theorem 1.32. Every closed subspace (A, τA ) of a compact topological
space (X, τ ) is compact.
Lemma 1.33. (The Tube Lemma) Let (X, τ ) and (Y, τ 0 ) be topological
spaces and suppose (Y, τ 0 ) is compact. If x0 ∈ X and N is an open set in
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X × Y which contains the slice {x0 } × Y then there exists a neighbourhood
W of x0 such that N contains the tube W × Y .
Theorem 1.34. Let (X, τ ) and (Y, τ 0 ) be compact topological spaces. Then
X × Y is a compact topological space (with respect to the product topology).
Theorem 1.35. A subset K of Rn is compact if and only if K is both closed
and bounded.
Corollary 1.36. (Extreme Value Theorem) Let (X, τ ) be a compact topological space and let f : X → R be a continuous function. Then there exists
u, v ∈ X such that f (u) ≤ f (x) ≤ f (v) for all x ∈ X.
Using the fact that the unit sphere in a finite dimensional normed vector
space is compact we can prove:
Corollary 1.37. All norms on a finite dimensional vector space are equivalent.
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MA2223: TOPOLOGICAL PROPERTIES OF METRIC SPACES
1.6. Uniformly continuous mappings.
Definition 1.38. Let (X, d) and (Y, d0 ) be metric spaces. A mapping T :
X → Y is called uniformly continuous if given any > 0 there exists δ > 0
such that
d(x1 , x2 ) < δ =⇒ d0 (T (x1 ), T (x2 )) < The difference between a uniformly continuous mapping and a continuous mapping ([MA2223 Metric spaces: Definition 1.35]) is that uniform
continuity is a global condition while continuity is local.
Lemma 1.39. (Lebesgue Lemma) Let (X, d) be a compact metric space. If
{Uα } is an open cover of X then there exists δ > 0 such that each subset A
of X with diameter less than δ is contained in an element of {Uα }. (δ is
called a Lebesgue number for the open cover).
Theorem 1.40. (Uniform continuity theorem) Let (X, d) and (Y, d0 ) be
metric spaces and T : X → Y a continuous mapping. If (X, d) is compact
then T is uniformly continuous.
A uniformly continuous map is continuous, but the converse is not true.
Example 1.41.
(i) f : R\{0} → R, x 7→
1
x
is continuous but not uni-
formly continuous.
(ii) f : R → R, x 7→ x2 is continuous but not uniformly continuous.
(iii) Contractions on a metric space are uniformly continuous.
(iv) Isometries between metric spaces are uniformly continuous.
(v) Continuous linear operators between normed vector spaces are uniformly continuous.
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1.7. Sequentially compact and totally bounded spaces.
Definition 1.42. A topological space (X, τ ) is sequentially compact if every
∞
sequence (xn )∞
n=1 in X has a subsequence (xnj )j=1 which converges to a point
in X.
Definition 1.43. A metric space (X, d) is called totally bounded if given
any r > 0 there exists a finite collection of open balls B(x1 , r), . . . , B(xn , r)
of radius r which covers X.
Theorem 1.44. Let (X, d) be a metric space. The following statements are
equivalent:
(i) (X, d) is compact,
(ii) (X, d) is sequentially compact,
(iii) (X, d) is complete and totally bounded.
Corollary 1.45. (Bolzano-Weierstrass Theorem) Every bounded sequence
in Rn has a convergent subsequence.
We saw that in Rn a set is compact if and only if it is closed and
bounded. This fact does not extend to infinite dimensional spaces. For
example, the unit sphere in the sequence space `2 is closed and bounded
but not totally bounded and hence not compact. It is still possible to obtain analogous results for infinite dimensional spaces. In the case of the
Banach spaces (C[a, b], k.k∞ ) this is achieved by introducing “equicontinuity” and leads to a version of the Bolzano-Weierstrass theorem for sequences
in (C[a, b], k.k∞ ). For more on this see the Arzela-Ascoli theorem.