Support of a function

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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.


[Ru] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 38\, .
How to Cite This Entry:
Support of a function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article