Support of a Real Function on a Metric Space

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $(X,d)$ be a metric space and let $f:X\to\mathbb{R}$ be a function.

(i) Support. The support of $f$ is the closure

$$\operatorname{supp}f=\operatorname{cl}\{x\in X:f(x)\neq0\}.$$

(ii) Compact support. $f$ has compact support if $\operatorname{supp}f$ is compact.