Preclosure operator

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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. (Preservation of nullary unions);
  2. (Extensivity);
  3. (Preservation of binary unions).

The last axiom implies the following:

4. implies .

Topology[edit]

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 .

Examples[edit]

Premetrics[edit]

Given a premetric on , then

is a preclosure on .

Sequential spaces[edit]

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 .

See also[edit]

References[edit]