support of function

DefinitionSuppose $X$ is a topological space, and $f\\colon X\\to\\mathbb{C}$ is a function.Then the support of $f$ (written as $\\operatorname{supp}f$ ), is the set

\\operatorname{supp}f=\\overline{\\{x\\in X\\mid f(x)\eq 0\\}}.

In other words, $\\operatorname{supp}f$ is the closure of the set where $f$ does not vanish.

Properties

Let $f\\colon X\\to\\mathbb{C}$ be a function.

1.
$\\operatorname{supp}f$ is closed.
2.
If $x\otin\\operatorname{supp}f$ , then $f(x)=0$ .
3.
If $\\operatorname{supp}f=\\emptyset$ , then $f=0$ .
4.
If $\\chi\\colon X\\to\\mathbb{C}$ is such that $\\chi=1$ on $\\operatorname{supp}f$ , then $f=\\chi f$ .
5.
If $f,g\\colon X\\to\\mathbb{C}$ are functions, then we have
$\\displaystyle\\operatorname{supp}(fg)$ $\\displaystyle\\subset$ $\\displaystyle\\operatorname{supp}f\\cap\\operatorname{supp}g,$
$\\displaystyle\\operatorname{supp}(f+g)$ $\\displaystyle\\subset$ $\\displaystyle\\operatorname{supp}f\\cup\\operatorname{supp}g.$
6.
If $Y$ is another topological space, and $\\Psi\\colon Y\\to X$ is ahomeomorphism, then
$\\operatorname{supp}(f\\circ\\Psi)=\\Psi^{-1}(\\operatorname{supp}f).$