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 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 .
Topology 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;Eduard Čech, Zdeněk Frolík, Miroslav Katětov, Topological spaces Prague: Academia, Publishing House of the Czechoslovak Academy of Sciences, 1966, Theorem 14 A.9 . however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.S. Dolecki, An Initiation into Convergence Theory, in F. Mynard, E. Pearl (editors), Beyond Topology, AMS, Contemporary Mathematics, 2009.
Examples Premetrics Given a premetric on , then
is a preclosure on
Sequential spaces 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 Eduard Čech
References
A.V. Arkhangelskii, L.S. Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. . B. Banaschewski, Bourbaki's Fixpoint Lemma reconsidered, Comment. Math. Univ. Carolinae 33 (1992), 303–309.