Definition:Interior (Topology)

Definition

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

Let $H \subseteq T$.

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:

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

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

Alternative Definition

The following definition for interior is equivalent to the above:

• $H^\circ$ is the largest open set contained in $H$.

This fact is demonstrated in Set Interior is Largest Open Set.

Interior Point

An interior point of $H$ is any point in the interior of $H$.

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 this website, $H^\circ$ is the notation of choice.

Also see

• Closure

• Results about Set Interiors can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Closures and Interiors