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 where is the power set of X. The preclosure operator has to satisfy the following properties: The last axiom implies the following:

Topology

A set A is closed (with respect to the preclosure) if [A]_p=A. A set U \subset X is open (with respect to the preclosure) if its complement is closed. The collection of all open sets generated by the preclosure operator is a topology; however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.

Examples

Premetrics

Given d a premetric on X, then is a preclosure on X.

Sequential spaces

The sequential closure operator is a preclosure operator. Given a topology \mathcal{T} 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