closure axioms


A closure operatorPlanetmathPlanetmathPlanetmath on a set X is an operator which assigns a set Ac to each subset A of X, and such that the following (Kuratowski’s closure axioms) hold for any subsets A and B of X:

  1. 1.

    c=;

  2. 2.

    AAc;

  3. 3.

    (Ac)c=Ac;

  4. 4.

    (AB)c=AcBc.

The following theorem due to Kuratowski says that a closure operator characterizes a unique topologyMathworldPlanetmath on X:

Theorem. Let c be a closure operator on X, and let 𝒯={X-A:AX,Ac=A}. Then 𝒯 is a topology on X, and Ac is the 𝒯-closureMathworldPlanetmath of A for each subset A of X.

Title closure axioms
Canonical name ClosureAxioms
Date of creation 2013-03-22 13:13:44
Last modified on 2013-03-22 13:13:44
Owner Koro (127)
Last modified by Koro (127)
Numerical id 9
Author Koro (127)
Entry type Definition
Classification msc 54A05
Synonym Kuratowski’s closure axioms
Synonym Kuratowski closure axioms
Related topic Closure
Defines closure operator