# Definition:Interior (Topology)

## Definition

Let $T = \struct {S, \tau}$ 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:

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

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

### Definition 2

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

### Definition 3

The interior of $H$ is the set of all interior points of $H$.

## Notation

The interior of $H$ can be denoted:

$\map {\mathrm {Int} } H$
$H^\circ$

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.

## Examples

### Closed Real Interval in Closed Unbounded Real Interval

Let $\R$ be the real number line under the usual (Euclidean) metric.

Let $M$ be the subspace of $\R$ defined as:

$M = \hointl \gets b$

Let $S$ be the closed real interval defined as:

$S = \closedint a b$

Then the interior of $S$ in $M$ is given by:

$S^\circ = \hointl a b$

## Also see

• Results about set interiors can be found here.