closure axioms

A closure operator on a set $X$ is an operator which assigns a set $A^{c}$ 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.

$\emptyset^{c}=\emptyset$ ;
2.

$A\subset A^{c}$ ;
3.

$(A^{c})^{c}=A^{c}$ ;
4.

$(A\cup B)^{c}=A^{c}\cup B^{c}.$

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

Theorem. Let $c$ be a closure operator on $X$ , and let $\mathcal{T}=\{X-A:A\subseteq X,\;A^{c}=A\}$ . Then $\mathcal{T}$ is a topology on $X$ , and $A^{c}$ is the $\mathcal{T}$ -closure 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