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:
- ;
- ;
- .
The last axiom implies the following:Topology
A set is closed if. A set is open if is closed. The collection of all open sets generated by the preclosure operator is a pretopology.Examples
Premetrics
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.
