# 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. 1.

$\operatorname{supp}f$ is closed.

2. 2.

If $x\notin\operatorname{supp}f$, then $f(x)=0$.

3. 3.

If $\operatorname{supp}f=\emptyset$, then $f=0$.

4. 4.

If $\chi\colon X\to\mathbb{C}$ is such that $\chi=1$ on $\operatorname{supp}f$, then $f=\chi f$.

5. 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. 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 SupportOfFunction 2013-03-22 13:46:10 2013-03-22 13:46:10 matte (1858) matte (1858) 16 matte (1858) Definition msc 54-00 support carrier ZeroOfAFunction ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces