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.
${\mathrm{\varnothing}}^{c}=\mathrm{\varnothing}$;

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 $\mathcal{T}=\{XA:A\subseteq X,{A}^{c}=A\}$. 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  20130322 13:13:44 
Last modified on  20130322 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 