↑ arvau

Wikipedia / 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

is the power set

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:

  1. .

Topology

is closed 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

a premetric, then

Sequential spaces

The sequential closure with respect to which the sequential closure operator is defined, the topological space is a sequential space

See also

References