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.
A preclosure operator on a set is a map
where is the power set of .
The preclosure operator has to satisfy the following properties:
- (Preservation of nullary unions);
- (Preservation of binary unions).
The last axiom implies the following:
- 4. implies .
A set is closed (with respect to the preclosure) if . A set is open (with respect to the preclosure) if is closed. The collection of all open sets generated by the preclosure operator is a pretopology.
Given a premetric on , then
is a preclosure on .