Locally Compact Metric Space

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $(X,d)$ be a metric space.

(i) Locally compact metric space. $(X,d)$ is locally compact if for every $x\in X$ there is a real number $r>0$ such that the closed ball $\bar{B}(x,r)$ is compact.