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 $X$ is a map $$quadp

:$$quadp:\mathcal(X) \to \mathcal(X)

where $\mathcal(X)$ is the power set of $X$.

The preclosure operator has to satisfy the following properties:
# varnothingp = \varnothing \! (Preservation of nullary unions);
# A \subseteq p (Extensivity);
# \cup Bp = p \cup p (Preservation of binary unions).

The last axiom implies the following:

: 4. $A \subseteq B$ implies p \subseteq p.

Topology

A set $A$ is closed (with respect to the preclosure) if p=A. A set $U\subset X$ is open (with respect to the preclosure) if $A=X\setminus U$ is closed. The collection of all open sets generated by the preclosure operator is a topology; 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 $d$ a premetric on $X$, then

: p=\

is a preclosure on $X$.

Sequential spaces

The sequential closure operator $$quad\text is a preclosure operator. Given a topology $\mathcal$ with respect to which the sequential closure operator is defined, the topological space $(X,\mathcal)$ is a sequential space if and only if the topology $\mathcal_\text$ generated by $$quad\text is equal to $\mathcal$, that is, if $\mathcal_\text=\mathcal$.

See also

* Eduard Čech

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.

Closure operators