# Definition:Support of Distribution

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## Definition

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

This definition needs to be completed.In particular: The link goes to a definition of an open set which has been made only for the real number line. Needs to be expanded to include the general real cartesian space. This has already been defined in the context of Metric Spaces; it's just a matter of creating a page which connects the concepts of an open set on a real number line to that on a general space.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{DefinitionWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Let $\map \DD \Omega$ be the space of continuous functions compactly supported in $\Omega$.

Let $\map {\DD'} \Omega$ be the distribution space.

Let $T \in \map {\DD'} \Omega$ be a distribution.

The **support** $\map {\mathrm {supp} } T \subseteq \Omega$ of $T$ is defined by:

- $\ds x \notin \map {\mathrm {supp} } T$ if and only if:
- there exists an open neighborhood $U$ of $x$ such that:
- for all $\phi \in \map \DD \Omega$ such that $\map {\mathrm {supp} } \phi \subseteq U$:
- $\map T \phi = 0$

- for all $\phi \in \map \DD \Omega$ such that $\map {\mathrm {supp} } \phi \subseteq U$:

- there exists an open neighborhood $U$ of $x$ such that: