A function$f\colon X \to V$ on a topological space with values in a vector space$V$ (or really any pointed set with the basepoint called $0$) has compact support (or is compactly supported) if the closure of its support, the set of points where it is non-zero, is a compact subset. That is, the subset $\overline{f^{-1}(V \setminus \{0\})}$ is a compact subset of $X$.

Typically, $X$ is Hausdorff, $f$ is a continuous function, and $V$ is a Hausdorff topological vector space (or at least a pointed topological space whose basepoint is closed), so that $f^{-1}(V \setminus \{0\})$ is an open subspace of $X$, yet any compact subspace of $X$ must be closed; this is why we take the closure.

If we work with locales instead of topological spaces, then a closed point $0$ in $V$ still has an open subspace of $V$ as its formal dual, and we use this in the place of $V \setminus \{0\}$.