support of functionSupportOfFunction2003-07-18 11:14:492006-10-21 14:44:43Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
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\newcommand{\supp}[0]{\operatorname{supp}}
{\bf Definition}
Suppose $X$ is a topological space, and $f\colon X\to \sC$ is a function.
Then the \emph{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.
\subsubsection*{Properties}
Let $f\colon X\to \sC$ be a function.
\begin{enumerate}
\item $\supp f$ is closed.
\item If $x\notin \supp f$, then $f(x)=0$.
\item If $\supp f = \emptyset$, then $f=0$.
\item If $\chi\colon X\to \sC$ is such that $\chi = 1$ on $\supp f$, then
$f=\chi f$.
\item 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*}
\item If $Y$ is another topological space, and $\Psi\colon Y\to X$ is a
homeomorphism, then
$$
\supp (f\circ \Psi) = \Psi^{-1}(\supp f).
$$
\end{enumerate}