Definition:Exterior (Topology)

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:

$\map {\mathrm {Ext} } H$

$H^e$

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

• Equivalence of Definitions of Exterior

• Definition:Interior (Topology)

• Results about set exteriors can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exterior (of a set)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exterior (of a set)