interior axioms

Let S be a set. Then an interior operator is a function :𝒫(S)𝒫(S) which satisfies the following properties:

Axiom 1.


Axiom 2.

For all XS, one has XS.

Axiom 3.

For all XS, one has (X)=X.

Axiom 4.

For all X,YS, one has (XY)=XY.

If S is a topological spaceMathworldPlanetmath, then the operator which assigns to each set its interior satisfies these axioms. Conversely, given an interior operator on a set S, the set {XXS} defines a topology on S in which X is the interior of X for any subset X of S. Thus, specifying an interior operator on a set is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to specifying a topology on that set.

The concepts of interior operator and closure operatorPlanetmathPlanetmathPlanetmath are closely related. Given an interior operator , one can define a closure operator c by the condition


and, given a closure operator c, one can define an interior operator by the condition

Title interior axioms
Canonical name InteriorAxioms
Date of creation 2013-03-22 16:30:37
Last modified on 2013-03-22 16:30:37
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Definition
Classification msc 54A05
Related topic GaloisConnection
Defines interior operator