Support of a function

2020 Mathematics Subject Classification: Primary: 54A [MSN][ZBL]

Let $X$ be a topological space and $f:X\to \mathbb R$ a function. The support of $f$, denoted by ${\rm supp}\, (f)$ is the smallest closed set outside of which the function $f$ vanishes identically. ${\rm supp}\, (f)$ can also be characterized as

• the complent of the union of all sets on which $f$ vanishes identically
• the closure of the set $\{f\neq 0\}$.

The same concept can be readily extended to maps taking values in a vector space or more generally in an additive group.

A function $f$ is said to have compact support if ${\rm supp}\, (f)$ is compact. If the target $V$ is a vector space, the set of functions $f:X\to V$ with compact support is also a vector space.

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