View source for Closure space

Let $S$ be a set and $\mathcal{P}S$ the set of subsets of $S$. A function $K : \mathcal{P}S \rightarrow \mathcal{P}S$ is a ''closure operation'' if for all $A,B \in \mathcal{P}S$:

K1) $A \subseteq K(A)$;

K2) $K(K(A)) \subseteq K(A)$;

K3) $K(A) \subseteq K(A\cup B)$;

K4) $K(A\cup B) \subseteq K(A) \cup K(B)$;

K5) $K(\emptyset) = \emptyset$. 

These are the ''Kuratowski closure axioms'', and a function $K$ satisfying these axioms is called a ''Kuratowski closure operator'' (or Kuratowski closure operation).

A pair $(S,\mathrm{Cl})$ is a ''closure space'' if and only if $\mathrm{Cl}$ satisfies:
$$
A \subseteq \mathrm{Cl}(B) \Leftrightarrow \mathrm{Cl}(A) \subseteq \mathrm{Cl}(B) \ .
$$

This condition is equivalent to K1)–K3). A closure space that satisfies K5) is a Fréchet $V$-space. A Fréchet $V$-space that satisfies K4) is a [[topological space]].

A Čech closure operator is a mapping $C : \mathcal{P}S \rightarrow \mathcal{P}S$ such that

C1) $C(\emptyset) = \emptyset$;

C2) (${=}$ K1)) $A \subseteq C(A)$;

C3) $C(A \cup B) = C(A) \cup C(B)$. 

A set endowed with a Čech closure operator is called a ''pre-topological space''.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N.M. Martin,   S. Pollard,   "Closure spaces and logic" , Kluwer Acad. Publ.  (1996)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , Graduate Texts in Mathematics '''27''' Springer  (1975) ISBN 0-387-90125-6  {{ZBL|0306.54002}}</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Dikranjan,   W. Tholin,   "Categorical structures of closure operators" , Kluwer Acad. Publ.  (1996)</TD></TR>
</table>

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