Definition:Support of Distribution

Definition
Let $\Omega \subseteq \R^n$ be an open set.

Let $\mathcal D \left({\Omega}\right)$ be the space of continuous functions compactly supported in $\Omega$.

Let $T \in \mathcal D\,' \left({\Omega}\right)$ be a distribution.

The support $\operatorname{supp} \left({T}\right) \subseteq \Omega$ of $T$ is defined by:
 * $\displaystyle x \notin \operatorname{supp} \left({T}\right)$ :
 * there exists an open neighborhood $U$ of $x$ such that:
 * for all $\phi \in \mathcal D \left({\Omega}\right)$ such that $\operatorname{supp} \left({\phi}\right) \subseteq U$:
 * $T \left({\phi}\right) = 0$