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 topology. However, the topological closure of a set in this topology will not necessarily be the same as .
The closure operator cl on this topological space satisfies for all .
Given a premetric on , then
is a preclosure on .
The sequential closure operator is a preclosure operator. Given a topology with respect to which the sequential closure operator is defined, the topological space is a sequential space if and only if the topology generated by is equal to , that is, if .