nLab interior

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:

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

Related entries

• interior point

• ionad

Reference

• M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (exposé IV, exercise 9.4.8, pp.461-462)