Preclosure operator


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 is a map
where is the power set of
The preclosure operator has to satisfy the following properties:
  1. ;
  2. ;
  3. .
The last axiom implies the following:

Topology

A set is closed if. A set is open 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 a premetric on, then
is a preclosure on

Sequential spaces

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