# Definition:Exterior (Topology)

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## Definition

Let $T$ be a topological space.

Let $H \subseteq T$.

### Definition 1

The **exterior** of $H$ is the complement of the closure of $H$ in $T$.

### Definition 2

The **exterior** of $H$ is the interior of the complement of $H$ in $T$.

## Notation

The exterior of $H$ can be denoted:

- $\operatorname{Ext} \left({H}\right)$
- $H^e$

The first is regarded by some as cumbersome, but has the advantage of being clear.

$H^e$ is neat and compact, but has the disadvantage of being relatively obscure.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the notation of choice is $H^e$.

## Also see

- Results about
**set exteriors**can be found here.