nLab interior

Context

Topology

topology

algebraic topology

Contents

Definition

For $S \subset X$ a subset of a topological space $X$, a, interior point of $S$ is a point $x \in S$ which has a neighbourhood in $X$ that is contained in $S$. The union of all interior points is the interior $S^\circ$ of $S$. It can be defined as the largest open set contained in $S$.

In general, we have $S^\circ \subseteq S$. $S$ is open if and only if $S^\circ = S$, so that in particular $S^{\circ\circ} = S^\circ$. This makes the interior operator $P(X) \to P(X): S \mapsto S^\circ$ a co-closure operator. It also satisfies the equations $(S \cap T)^\circ = S^\circ \cap T^\circ$ and $X^\circ = X$. Moreover, any co-closure operator $c$ on $P(X)$ that preserves finite intersections must be the interior operation for some topology, namely the family consisting of fixed points of $c$; this gives one of many equivalent ways to define a topological space.

Compare the topological closure $\bar{S}$ and frontier $\partial S = \bar{S} \setminus S^\circ$.

Remark

The interior of a subtopos $\mathcal{E}_j$ of a Grothendieck topos $\mathcal{E}$, as well as the exterior, were defined in an exercise in SGA4: $Int(\mathcal{E}_j)$ as the largest open subtopos contained in $\mathcal{E}_j$. The boundary of a subtopos is then naturally defined as the subtopos complementary to the (open) join of the exterior and interior subtoposes in the lattice of subtoposes.

Reference

Revised on August 19, 2015 04:47:11 by Thomas Holder (82.113.121.95)