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
- Results about set interiors can be found here.
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
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.5$: Topological spaces
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): frontier (boundary)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): frontier (boundary)