support of function

Definition Suppose $X$ 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)\neq 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 $x\notin\operatorname{supp}f$ , 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 a homeomorphism, then

$\operatorname{supp}(f\circ\Psi)=\Psi^{-1}(\operatorname{supp}f).$

Title	support of function
Canonical name	SupportOfFunction
Date of creation	2013-03-22 13:46:10
Last modified on	2013-03-22 13:46:10
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	16
Author	matte (1858)
Entry type	Definition
Classification	msc 54-00
Synonym	support
Synonym	carrier
Related topic	ZeroOfAFunction
Related topic	ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

	$\displaystyle\operatorname{supp}(fg)$	$\displaystyle\subset$	$\displaystyle\operatorname{supp}f\cap\operatorname{supp}g,$
	$\displaystyle\operatorname{supp}(f+g)$	$\displaystyle\subset$	$\displaystyle\operatorname{supp}f\cup\operatorname{supp}g.$