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Concept
Preclosure operator
Summary
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.
A preclosure operator on a set is a map
where is the power set of
The preclosure operator has to satisfy the following properties:
(Preservation of nullary unions);
(Extensivity);
(Preservation of binary unions).
The last axiom implies the following:
4. implies .
A set is closed (with respect to the preclosure) if . A set 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.
Given a premetric on , then
is a preclosure on
The sequential closure operator is a preclosure operator.
Official source
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