# Preclosure operator

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
is closed. The collection of all open sets generated by the preclosure operator is a pretopology.

## Examples

### 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

## References

- 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.