- Preclosure operator
In
topology , a preclosure operator, or Čech closure operator is a map between subsets of a set, similar to a topologicalclosure operator , except that it is not required to beidempotent . That is, a preclosure operator obeys only three of the fourKuratowski 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 is closed. The collection of all open sets generated by the preclosure operator is a topology.
The
closure operator cl on this topological space satisfies for all .Examples
Premetrics
Given a
prametric on , then:
is a preclosure on .
equential 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 asequential space if and only if the topology generated by is equal to , that is, if .ee also
*
Eduard Čech References
* A.V. Arkhangelskii, L.S.Pontryagin, "General Topology I", (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.
* B. Banascheski, [http://www.emis.de/journals/CMUC/pdf/cmuc9202/banas.pdf "Bourbaki's Fixpoint Lemma reconsidered"] , Comment. Math. Univ. Carolinae 33 (1992), 303-309.
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