Samiksha Jaiswal (Editor)

Preclosure operator

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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.

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Definition

A preclosure operator on a set X is a map [ ] p

[ ] p : P ( X ) → P ( X )

where P ( X ) is the power set of X .

The preclosure operator has to satisfy the following properties:

  1. [ ∅ ] p = ∅ (Preservation of nullary unions);
  2. A ⊆ [ A ] p (Extensivity);
  3. [ A ∪ B ] p = [ A ] p ∪ [ B ] p (Preservation of binary unions).

The last axiom implies the following:

4. A ⊆ B implies [ A ] p ⊆ [ B ] p .

Topology

A set A is closed (with respect to the preclosure) if [ A ] p = A . A set U ⊂ X is open (with respect to the preclosure) if A = X ∖ U is closed. The collection of all open sets generated by the preclosure operator is a topology. However, the topological closure of a set X in this topology will not necessarily be the same as [ S ] p .

The closure operator cl on this topological space satisfies [ A ] p ⊆ cl ⁡ ( A ) for all A ⊂ X .

Premetrics

Given d a premetric on X , then

[ A ] p = { x ∈ X : d ( x , A ) = 0 }

is a preclosure on X .

Sequential spaces

The sequential closure operator [ ] seq is a preclosure operator. Given a topology T with respect to which the sequential closure operator is defined, the topological space ( X , T ) is a sequential space if and only if the topology T seq generated by [ ] seq is equal to T , that is, if T seq = T .

References

Preclosure operator Wikipedia
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