Definition Suppose $X$ is a topological space, and $f\colon X\to \sC$ is a function. Then the support of $f$ (written as $\supp f$ ), is the set $$ \supp f = \overline{\{x\in X\mid f(x)\neq 0\}}. $$ In other words, $\supp f$ is the closure of the set where $f$ does not vanish.

Properties

1. $\supp f$ is closed.
2. If $x\notin \supp f$ , then $f(x)=0$ .
3. If $\supp f = \emptyset$ , then $f=0$ .
4. If $\chi\colon X\to \sC$ is such that $\chi = 1$ on $\supp f$ , then $f=\chi f$ .
5. If $f,g\colon X\to \sC$ are functions, then we have \begin{eqnarray*} \supp (fg) &\subset & \supp f \cap \supp g, \\ \supp (f+g) &\subset & \supp f \cup \supp g. \end{eqnarray*}
6. If $Y$ is another topological space, and $\Psi\colon Y\to X$ is a homeomorphism, then $$ \supp (f\circ \Psi) = \Psi^{-1}(\supp f). $$