# Definition:Interior (Topology)

From ProofWiki

*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.