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

[a1]	N.M. Martin, S. Pollard, "Closure spaces and logic" , Kluwer Acad. Publ. (1996)
[a2]	J.L. Kelley, "General topology" , Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002
[a3]	D. Dikranjan, W. Tholin, "Categorical structures of closure operators" , Kluwer Acad. Publ. (1996)