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

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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

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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;[1] however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.[2]

Examples

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Premetrics

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Given   a premetric on  , then

 

is a preclosure on  

Sequential spaces

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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

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References

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  1. ^ Eduard Čech, Zdeněk Frolík, Miroslav Katětov, Topological spaces Prague: Academia, Publishing House of the Czechoslovak Academy of Sciences, 1966, Theorem 14 A.9 [1].
  2. ^ S. Dolecki, An Initiation into Convergence Theory, in F. Mynard, E. Pearl (editors), Beyond Topology, AMS, Contemporary Mathematics, 2009.
  • A.V. Arkhangelskii, L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.
  • B. Banascheski, Bourbaki's Fixpoint Lemma reconsidered, Comment. Math. Univ. Carolinae 33 (1992), 303–309.