Proper metric space

Definition

A metric space $(X, d)$ is proper when each closed ball $\bar{B}(x, r)$, with $x \in X$ and $r \ge 0$, is compact.

Remarks

In a proper metric space, a subset is compact if and only if it is closed and bounded. Thus the Heine–Borel theorem says that $\mathbb{R}^n$ is proper. Each proper metric space is complete and locally compact. An infinite set with the discrete metric is not proper: the closed ball of radius $1$ is the whole set, which is not compact.

Explanations

L1 human

A proper metric space is a space where the Heine–Borel test for compactness works: closed and bounded is enough. Euclidean spaces are proper, but many infinite-dimensional spaces are not.