Metric spaces as topological spaces. Equivalent metrics. Metrization problem. Isometric metric spaces.

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Metric spaces. A metric space is a kind of topological space. In a metric space any union of open sets is open and any finite intersection of open sets is open. Consequently a metric space meets the axiomatic requirements of a topological space and is thus a topological space. It was, in fact, this particular property of a metric space that was used to define a topological space.

Theorem 1. The collection of open spheres in a set X with metric d is a base for a topology on X.

Def. Metric topology. Let d be a metric on a non-empty set X. The topology τ on X generated by the collection of open spheres in X is called the metric topology (or, the topology induced by the metric d).

The set X together with the topology τ induced by the metric d is a metric space. A metric space then can be viewed as a topological space in which the topology is induced by a metric.

Example 1. Let d be the usual metric in three dimensional space R³. Then the set of open spheres in R³ constitute a base for a topology on R³. Thus the usual metric on R³ induces the usual topology on R³, the collection of all open sets.

Example 2. Let d be the usual metric on the real line R, i.e. d(a, b) = |a - b|. Then the open spheres in R correspond to the finite open intervals in R. Thus the usual metric on R induces the usual topology, the set of all open intervals, on R.

Example 3. Let d be the trivial metric on some set X. Note that for any p ε X, S(p, 1/2) = {p}. Thus every singleton set (set consisting of only one element) is open and consequently every set is open. Hence the trivial metric induces the discrete topology on X.

Properties of metric topologies

Theorem 2. Let p be a point is a metric space X. Then the countable class of open spheres {S(p, 1), S(p, 1/2), S(p, 1/3), S(p, 1/4), .....}is a local base at p.

Theorem 3. The closure of a subset A of a metric space X is the set of points whose distance from A is zero.

In a metric space all singleton sets {p} are closed.

Theorem 4. In a metric space all finite sets are closed.

Following is an important “separation” property of metric spaces.

Theorem 5 (Separation Axiom). Let A and B be closed disjoint subsets of a metric space. Then there exist disjoint open sets G and H such that A G and B H. See Fig. 1.

One might think that the distance between two disjoint closed sets would be greater than zero. However, this is not necessarily the case as the following example shows.

Example 4. The two sets

A = {(x, y): xy -1, x < 0}

B = {(x, y): xy 1, x > 0}

are shown in Fig. 2. Both sets are closed and they are disjoint. However, d(A, B) = 0.

Def. Equivalent metrics. Two metrics on a set X are said to be equivalent if and only if they induce the same topology on X.

Two metrics d and d* on a set X induce the same topology on X if and only if the open spheres of the d metric and the open spheres of the d* metric are bases for the same topology on X.

Example 5. Let P(x₁, y₁) and Q(x₂, y₂) be two arbitrary points in the plane R². The usual metric d and the two metrics d₁ and d₂ defined by

d₁(P, Q) = max ( |x₁ - x₂|, |y₁ - y₂| )

and

d₂(P, Q) = |x₁ - x₂| + |y₁ - y₂|

all induce the usual topology on the plane R² since the collection of open spheres of each metric is a base for the usual topology on R². Fig. 3 shows the open spheres of each of the three metrics. Thus the three metrics are equivalent.

Metrization problem. Given any topological space X with a topology τ it is natural to ask if a metric exists for the space i.e. if there exists any metric d that would induce that topology τ. If one exists the space is said to be metrizable. Under what conditions is a particular topological space metrizable?

A topological space is metrizable if a distance between points can be defined so the space is a metric space such that open sets in the original space are open sets in the metric space, and conversely; i.e. there exists a topological transformation between the given space and a metric space. A compact Hausdorff is metrizable if and only if it satisfies the second axiom of countability; a regular T₁ topological space is metrizible if it satisfies the second axiom of countability (Urysohn’s theorem). A topological space is metrizible if and only if it is a regular T₁ topological space whose topology has a base B which is the union of a countable number of classes {B_n} of open sets which have the property that, for each point x and class B_n, there is a neighborhood of x which intersects only a finite number of the members of B_n.

James & James. Mathematics Dictionary

Isometric metric spaces. A metric space (X, d) is isometric to a metric space (Y, e) if and only if there exists a one-to-one onto function f: X Y which preserves distances, i.e. for all p, q ε X,

d(p, q) = e(f(p), f(q))

Theorem 6. If the metric space (X, d) is isometric to (Y, e), then (X, d) is also homeomorphic to (Y, e).

References

1. Lipschutz. General Topology

2. Simmons. Introduction to Topology and Modern Analysis

3. James & James. Mathematics Dictionary