Definition:Interior (Topology)
This page is about Interior in the context of topology. For other uses, see Interior.
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.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): interior (of a set)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): interior (of a set)