# Definition:Interior (Topology)/Definition 1

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## Contents

## Definition

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

Let $H \subseteq S$.

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

## Notation

The **interior** of $H$ can be denoted:

- $\map {\operatorname {Int} } H$
- $H^\circ$

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.

## Also see

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 1$: Closures and Interiors - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.7$: Definitions: Definition $3.7.24$