Support of a function

Definition

Let $f:X\to\mathbb F$ be a function on a topological space, where $\mathbb F$ has a distinguished zero element. The support of $f$ is

\operatorname{supp}(f)=\overline{\{x\in X:f(x)\ne0\}}.

Thus the support is a closed subset of $X$ , and $f$ vanishes on its complement.

Why the closure is included

The nonzero set need not be closed. Taking its closure includes boundary points at which $f$ may vanish but arbitrarily nearby values are nonzero. This convention makes compact support mean that this closed set is compact.

Uses

Families of supports appear in the local-finiteness condition for partitions of unity. Supports also localize test functions, sections, distributions, and other analytic objects, whose specialized notions of support extend this definition.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: supports, bump functions, and partitions of unity.