Definition:Kuratowski Closure Operator/Definition 1

From ProofWiki
Jump to: navigation, search

Definition

Let $S$ be a set.

Let $\operatorname {cl}:\mathcal P(S) \to \mathcal P(S)$ be a mapping from the power set of $S$ to itself.

Then $\operatorname{cl}$ is a Kuratowski closure operator iff it satisfies the following Kuratowski closure axioms for all $A, B \subseteq S$:

\((1):\)   \(\displaystyle A \subseteq \operatorname{cl}(A) \)             $\operatorname{cl}$ is inflationary
\((2):\)   \(\displaystyle \operatorname{cl} \left({ \operatorname{cl} (A) }\right) = \operatorname{cl} (A) \)             $\operatorname{cl}$ is idempotent
\((3):\)   \(\displaystyle \operatorname{cl} \left({A \cup B}\right) = \operatorname{cl}(A) \cup \operatorname{cl}(B) \)             $\operatorname{cl}$ preserves binary unions
\((4):\)   \(\displaystyle \operatorname{cl}(\varnothing) = \varnothing \)             


Note that axioms $(3)$ and $(4)$ may be replaced by the single axiom that for any finite subset, $\mathcal F$, of $\mathcal P(S)$:

$\displaystyle \operatorname{cl} \left({ \bigcup \mathcal F }\right) = \bigcup_{F \mathop \in \mathcal F} \left({ \operatorname{cl}(F) }\right)$