Definition:Closure Operator

From ProofWiki

Jump to: navigation, search

Definition

Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$.

A closure operator on $S$ is a mapping:

$\operatorname{cl}: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$

which satisfies the following conditions for all sets $X, Y \subseteq S$:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle X$	$\subseteq$	$\displaystyle \operatorname{cl} \left({X}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	i.e. $\operatorname{cl}$ is extensive
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle X \subseteq Y$	$\implies$	$\displaystyle \operatorname{cl} \left({X}\right) \subseteq \operatorname{cl} \left({Y}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	i.e. $\operatorname{cl}$ is increasing
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({X}\right)}\right)$	$=$	$\displaystyle \operatorname{cl} \left({X}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	i.e. $\operatorname{cl}$ is idempotent

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle X\)	\(\subseteq\)	\(\displaystyle \operatorname{cl} \left({X}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	i.e. $\operatorname{cl}$ is extensive
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle X \subseteq Y\)	\(\implies\)	\(\displaystyle \operatorname{cl} \left({X}\right) \subseteq \operatorname{cl} \left({Y}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	i.e. $\operatorname{cl}$ is increasing
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({X}\right)}\right)\)	\(=\)	\(\displaystyle \operatorname{cl} \left({X}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	i.e. $\operatorname{cl}$ is idempotent