Concept

Preclosure operator

Summary
In topology, a preclosure operator or Čech closure operator is a map between subsets of a set, similar to a topological closure operator, except that it is not required to be idempotent. That is, a preclosure operator obeys only three of the four Kuratowski closure axioms. Definition A preclosure operator on a set X is a map [\ \ ]_p :[\ \ ]_p:\mathcal{P}(X) \to \mathcal{P}(X) where \mathcal{P}(X) is the power set of X. The preclosure operator has to satisfy the following properties:

[\varnothing]_p = \varnothing ! (Preservation of nullary unions);

A \subseteq [A]_p (Extensivity);

[A \cup B]_p = [A]_p \cup [B]_p (Preservation of binary unions).

The last axiom implies the following: : 4. A \subseteq B implies [A]_p \subseteq [B]_p. Topology A set A is closed (with respect to the preclosure) if [A]_p
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