Definition:Exterior (Topology)
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Definition
Let $T$ be a topological space.
Let $H \subseteq T$.
The exterior of $H$ is the complement of the closure of $H$ in $T$.
Alternatively, 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 this website, $H^e$ is the notation of choice.
Also see
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Closures and Interiors
