Uniform local compactness

Question

In course of reading the paper “between Compactness and completeness” I came through the term uniform local compactness, which is not defined in the literature. I couldn’t find it in google. Would some please define the idea of uniform local compactness for me?

Answer 1

The explanation given in the cited paper (section 2), is that if $X$ is a metric space then define $ν(x) = \sup\{\epsilon > 0: S_\epsilon[x] \text{ is compact}\}$. Then $X$ is uniformly locally compact iff $\inf \{ν(x): x \in X\} > 0$. While the paper claims this is equivalent to some other unstated definition, rather than giving it as a definition, it is a very natural definition for the term "uniformly locally compact": not only is the space locally compact, but there is some global $\epsilon > 0$ such that the closed ball of radius $\epsilon$ around a point is always compact.

Answer 2

This article from the encyclopaedia of General Topology defines it as that $X$ has an entourage $U$ (for its uniformity) such that for all $x$, $U[x]$ is compact. It also gives $\omega_1$ as an example of a locally compact space which is not uniformly locally compact.