10.3 Metric spaces

中文

To generalize the idea of ‘distance’ as it appears in $\mathbb{R}$ and ${\mathbb{R}}^2$ , we define a metric space [9] to be a set $M$ with a distance function or metric $d:M\times M\to \mathbb{R}$ such that

(Met1) $d(x,y)\ge 0$ for all $x,y\in M.$

(Met2) $d(x,y)=0$ if and only if $x=y.$

(Met3) $d(x,y)=d(y,x)$

(Met4) $d(x,y)+d(y,z)\ge d(x,z).$

Condition (Met4) is called the triangle inequality – the length of any side of a triangle $xyz$ is less than the sum of the other two sides. For every $x$ in a metric space $(M,d)$ and positive real number $a>0$ we define the open ball $B_a(x)=\{y\mid d(x,y)<a\}$

In $n$-dimensional Euclidean space ${\mathbb{R}}^n$ the distance function is given by

$$d(\mathbf{x},\mathbf{y})=|\mathbf{x}-\mathbf{y}|=\sqrt{(x_1-y_1{)}^2+(x_2-y_2{)}^2+\cdots +(x_n-y_n{)}^2},$$

but the following could also serve as acceptable metrics:

$$d_1(\mathbf{x},\mathbf{y})=|x_1-y_1|+|x_2-y_2|+\cdots +|x_n-y_n|,$$ $$d_2(\mathbf{x},\mathbf{y})=\max \left(|x_1-y_1|,|x_2-y_2|,\dots ,|x_n-y_n|\right).$$

Exercise

Show that $d(\mathbf{x},\mathbf{y}),d_1(\mathbf{x},\mathbf{y})$ and $d_2(\mathbf{x},\mathbf{y})$ satisfy the metric axioms (Met1)–(Met4)

Exercise

In ${\mathbb{R}}^2$ sketch the open balls $B_1((0,0))$ for the metrics $d$, $d_1$ and $d_2$ .

If $(M,d)$ is a metric space, then a subset $U\subset M$ is said to be open if and only if for every $x\in U$ there exists an open ball $B_{ϵ}(x)\subseteq U$ . Just as for ${\mathbb{R}}^2$ , this defines a natural topology on $M$, called the metric topology. This topology is generated by the set of all open balls $B_a(x)\subset M$ . The proof closely follows the argument in Example 10.8.

In a metric space $(M,d)$ , a sequence $x_n$ converges to a point $x$ if and only if $d(x_n,x)\to 0$ as $n\to \infty$ . Equivalently, $x_n\to x$ if and only if for every $ϵ>0$ the sequence eventually enters and stays in the open ball $B_{ϵ}(x)$ . In a metric space the limit point $x$ of a sequence $x_n$ is unique, for if $x_n\to x$ and $x_n\to y$ then $d(x,y)\le d(x,x_n)+d(x_n,y)$ by the triangle inequality. By choosing $n$ large enough we have $d(x,y)<ϵ$ for any $ϵ>0$. . Hence $d(x,y)=0$, and $x=y$ by (Met2). For this reason, the concept of convergent sequences is more useful in metric spaces than in general topological spaces (see Problem 10.7).

In a metric space $(M,d)$ let $x_n$ be a sequence that converges to some point $x\in M$ . Then for every $ϵ>0$ there exists a positive integer $N$ such that $d(x_n,x_m)<ϵ$ for all $n,m>N$ For, let $N$ be an integer such that $d(x_k,x)<\frac{1}{2}ϵ$ for all $k>N$ , then

$$d(x_n,x_m)\le d(x_n,x)+d(x,x_m)<ϵ\text{ for all }n,m>N.$$

A sequence having this property, $d(x_n,x_m)\to 0$ as $n,m\to \infty$ , is termed a Cauchy sequence.

Example 10.11

Not every Cauchy sequence need converge to a point of $M$. For example, in the open interval (0, 1) with the usual metric topology, the sequence $x_n=2^{-n}$ is a Cauchy sequence yet it does not converge to any point in the open interval. A metric space $(M,d)$ is said to be complete if every Cauchy sequence $x_1,x_2,\dots$ . converges to a point $x\in M$ Completeness is not a topological property. For example the real line $\mathbb{R}$ is a complete metric space, and the Cauchy sequence $2^{-n}$ has the limit 0 in $\mathbb{R}$. The topological spaces $\mathbb{R}$ and (0, 1) are homeomorphic, using the map $\phi :x\mapsto \tan \frac{1}{2}\pi (2x-1)$ . However one space is complete while the other is not with respect to the metrics generating their topologies.

Problems

Problem 10.9

Show that every metric space is first countable. Hence show that every subset of a metric space can be written as the intersection of a countable collection of open sets.

Problem 10.10

If ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$ are two families of subsets of a set $X,$ show that the topologies generated by these families are homeomorphic if every member of ${\mathcal{U}}_2$ is a union of sets from ${\mathcal{U}}_1$ and vice versa. Use this property to show that the metric topologies on ${\mathbb{R}}^n$ defined by the metrics $d$, $d_1$ and $d_2$ are all homeomorphic.

Problem 10.11

A topological space $X$ is called normal if for every pair of disjoint closed subsets $A$ and $B$ there exist disjoint open sets $U$ and $V$ such that $A\subset U$ and $B\subset V$ . Show that every metric space is normal.