Definition 2

Let $X$ be a topological space and let $f$ be a real-valued function defined on $X$. The support of $f$, denoted by $\operatorname{Supp}(f)$, is the smallest closed set $S$ in $X$ such that $f(x) = 0$ for all $x \notin S$.

In other words, $\operatorname{Supp}(f)$ is the closure in $X$ of the set of all $x \in X$ such that $f(x) \neq 0$; or again, it is the set of all $x \in X$ such that every neighbourhood of $x$ contains a point $y$ for which $f(y) \neq 0$.

Let $(f_i)_{i \in I}$ be a family of finite real-valued functions on $X$ whose supports form a locally finite family; then the sum $\sum_{i \in I} f_i(x)$ is defined for each $x \in X$ (since it contains only a finite number of non-zero terms). The finite real-valued function $x \to \sum_{i \in I} f_i(x)$ is called the sum of the family $(f_i)$, and is denoted by $\sum_{i \in I} f_i$. If each of the $f_i$ is continuous, then so is $f = \sum_{i \in I} f_i$; for if $x$ is any point of $X$, there is a neighbourhood $V$ of $x$ which meets only a finite number of supports of the $f_i$, and hence there is a finite subset $H$ of $I$ such that $f(y) = \sum_{i \in H} f_i(y)$ for all $y \in V$.