距离空间

English draft

Translation of the statement above · machine-drafted, pending author review · the Chinese text is authoritative.

Abstract

A metric space is a pair $(X, d)$ with $d : X \times X \to [0, +\infty)$ satisfying $d(x,y) = 0 \iff x = y$, symmetry and the triangle inequality; open balls $B(x,r) = \{y : d(x,y) < r\}$ generate a topology, and they form a base for it. The point of the definition here is the split that follows: compactness and closedness use only the open sets, while completeness and total boundedness use $d$. Completeness is genuinely not topological — $\mathbb{R}$ and $(0,1)$ are homeomorphic but only the first is complete.

Statement

A metric space is a pair $(X, d)$: $X$ is a set and $d : X \times X \to [0, +\infty)$ is a metric on $X$, satisfying

• non-degeneracy: $d(x, y) = 0 \iff x = y$
• symmetry: $d(x, y) = d(y, x)$
• triangle inequality: $d(x, z) \le d(x, y) + d(y, z)$

From $d$ one defines the open balls $B(x, r) = \{ y \in X : d(x, y) < r \}$, and declares $U \subseteq X$ to be open $\iff U$ is a union of open balls. This puts a topology on $X$.

Given $\varepsilon > 0$, a set $D \subseteq X$ is called an $\varepsilon$-net if $\forall x \in X, \exists y \in D : d(x, y) < \varepsilon$.

Notes

The open balls form a base for this topology. Every metric space is first countable and normal (hence Hausdorff).

The properties that follow fall into two classes: compactness / closedness depend only on the open sets and are topological; completeness / total boundedness use $d$ and are metric. This dividing line is the point of the whole cluster.

⚠ Completeness is not a topological property: $\mathbb{R}$ and the open interval $(0, 1)$ are homeomorphic, yet only the first is complete.