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.


  2. 2.


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  4. 4.


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