PlanetMath: support of function

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support of function

(Definition)

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

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

In other words, $\operatorname{supp}f$ is the closure of the set where

does not vanish.

Properties

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

$\operatorname{supp}f$ is closed.
If $x\notin \operatorname{supp}f$ , then .
If $\operatorname{supp}f = \emptyset$ , then .
If $\chi\colon X\to \mathbb{C}$ is such that $\chi = 1$ on $\operatorname{supp}f$ , then $f=\chi f$ .
If $f,g\colon X\to \mathbb{C}$ are functions, then we have

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

$\displaystyle \operatorname{supp}(f+g)$ $\displaystyle \subset$ $\displaystyle \operatorname{supp}f \cup \operatorname{supp}g.$
If is another topological space, and $\Psi\colon Y\to X$ is a homeomorphism, then
$\displaystyle \operatorname{supp}(f\circ \Psi) = \Psi^{-1}(\operatorname{supp}f).$

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