# 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. 1.

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

2. 2.

$A\subset A^{c}$;

3. 3.

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

4. 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 ClosureAxioms 2013-03-22 13:13:44 2013-03-22 13:13:44 Koro (127) Koro (127) 9 Koro (127) Definition msc 54A05 Kuratowski’s closure axioms Kuratowski closure axioms Closure closure operator