# Definition:Interior (Topology)/Definition 1

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## Definition

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

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

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

## 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.

## Also see

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.24$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors