# Contents

## Definition

### For topological spaces

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 $\partial S = \bar{S} \setminus S^\circ$.

### For toposes

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.

## Properties

### Relation to topological closure

###### Lemma

Let $(X,\tau)$ be a topological space and let $S \subset X$ be a subset. Then the topological interior of $S$ equals the complement of the topological closure $Cl(X\backslash S)$ of the complement of $S$:

$Int(S) = X \backslash Cl\left( X \backslash S \right) \,.$
###### Proof

By taking complements once more, the statement is equivalent to

$X \backslash Int(S) = Cl( X \backslash S ) \,.$

Now we compute:

\begin{aligned} X \backslash Int(S) & = X \backslash \left( \underset{{U \, open} \atop {U \subset S}}{\cup}U \right) \\ & = \underset{U \subset S}{\cap} X \backslash U \\ & = \underset{{C\, closed} \atop {C \supset X \backslash S}}{\cap} C \\ & = Cl(X \backslash S) \end{aligned}

## Reference

Revised on May 9, 2017 13:11:41 by Urs Schreiber (217.92.118.241)