Definition:Interior (Topology)

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This page is about the interior of a topological space. For other uses, see Definition:Interior.

Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$.


Definition 1

The interior of $H$ is the union of all subsets of $H$ which are open in $T$.


That is, the interior of $H$ is defined as:

$\displaystyle H^\circ := \bigcup_{K \mathop \in \mathbb K} K$

where $\mathbb K = \left\{{K \in \tau: K \subseteq H}\right\}$.


Definition 2

The interior of $H$ is defined as the largest open set of $T$ which is contained in $H$.


Notation

The interior of $H$ can be denoted:

$\operatorname{Int} \left({H}\right)$
$H^\circ$

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

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


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


Also see

  • Results about set interiors can be found here.