metric space

A metric space is a set $X$ together with a real valued function $d: X \times X \longrightarrow \mathbb{R}$ (called a metric, or sometimes a distance function) such that, for every $x,y,z \in X$ ,

• $d(x,y) \geq 0$ , with equality ¹ if and only if $x=y$
• $d(x,y) = d(y,x)$
• $d(x,z) \leq d(x,y) + d(y,z)$

For $x \in X$ and $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$ , the open ball around $x$ of radius $\varepsilon$ is the set $B_\varepsilon(x) := \{y \in X \mid d(x,y) < \varepsilon\}$ . An open set in $X$ is a set which equals an arbitrary (possibly empty) union of open balls in $X$ , and $X$ together with these open sets forms a Hausdorff topological space. The topology on $X$ formed by these open sets is called the metric topology, and in fact the open sets form a basis for this topology (proof).

Similarly, the set $\bar{B}_\varepsilon(x) := \{y \in X \mid d(x,y) \leq \varepsilon\}$ is called a closed ball around $x$ of radius $\varepsilon$ . Every closed ball is a closed subset of $X$ in the metric topology.

The prototype example of a metric space is $\mathbb{R}$ itself, with the metric defined by $d(x,y) := |x-y|$ . More generally, any normed vector space has an underlying metric space structure; when the vector space is finite dimensional, the resulting metric space is isomorphic to Euclidean space.

Bibliography

1

J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.

Footnotes

... equality¹

This condition can be replaced with the weaker statement $d(x,y) = 0 \iff x=y$ without affecting the definition.