Axiom:Kuratowski Closure Axioms

Definition

The Kuratowski closure axioms are a set of conditions defining a closure operator allowing an alternative axiomatization of topological spaces.

Let $S$ and $T$ be sets.

$(\text K 1)$	$:$	$\ds \map \cl \empty = \empty $
$(\text K 2)$	$:$	$\ds S \subseteq \map \cl S $
$(\text K 3)$	$:$	$\ds \map \cl {\map \cl S} = \map \cl S $
$(\text K 4)$	$:$	$\ds \map \cl {S \cup T} = \map \cl S \cup \map \cl T $

1970: Stephen Willard: General Topology: Chapter $2$: Topological Spaces: $\S3$ Fundamental concepts

\((\text K 1)\)	$:$	\(\ds \map \cl \empty = \empty \)
\((\text K 2)\)	$:$	\(\ds S \subseteq \map \cl S \)
\((\text K 3)\)	$:$	\(\ds \map \cl {\map \cl S} = \map \cl S \)
\((\text K 4)\)	$:$	\(\ds \map \cl {S \cup T} = \map \cl S \cup \map \cl T \)