nLab
interior

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

For topological spaces

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

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

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

For toposes

The interior of a subtopos j\mathcal{E}_j of a Grothendieck topos \mathcal{E}, as well as the exterior, were defined in an exercise in SGA4: Int( j)Int(\mathcal{E}_j) as the largest open subtopos contained in j\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,τ)(X,\tau) be a topological space and let SXS \subset X be a subset. Then the topological interior of SS equals the complement of the topological closure Cl(X\S)Cl(X\backslash S) of the complement of SS:

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

By taking complements once more, the statement is equivalent to

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

Now we compute:

X\Int(S) =X\(UopenUSU) =USX\U =CclosedCX\SC =Cl(X\S) \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)