, 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);
- (Preservation of binary unions).
The last axiom implies the following:
- 4. implies .
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 .
is a preclosure on .
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
is equal to
, that is, if
- 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.