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 $[\ \ ]_{p}$

[\ \ ]_{p}:{\mathcal {P}}(X)\to {\mathcal {P}}(X)

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

The preclosure operator has to satisfy the following properties:

$[\varnothing ]_{p}=\varnothing \!$ (Preservation of nullary unions);
$A\subseteq [A]_{p}$ (Extensivity);
$[A\cup B]_{p}=[A]_{p}\cup [B]_{p}$ (Preservation of binary unions).

The last axiom implies the following:

4.

A\subseteq B

implies

[A]_{p}\subseteq [B]_{p}

.

Topology

A set $A$ is closed (with respect to the preclosure) if $[A]_{p}=A$ . A set $U\subset X$ is open (with respect to the preclosure) if its complement $A=X\setminus U$ is closed. The collection of all open sets generated by the preclosure operator is a topology;^[1] however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.^[2]

Examples

Premetrics

Given $d$ a premetric on $X$ , then

[A]_{p}=\{x\in X:d(x,A)=0\}

is a preclosure on $X.$

Sequential spaces

The sequential closure operator $[\ \ ]_{\text{seq}}$ is a preclosure operator. Given a topology ${\mathcal {T}}$ with respect to which the sequential closure operator is defined, the topological space $(X,{\mathcal {T}})$ is a sequential space if and only if the topology ${\mathcal {T}}_{\text{seq}}$ generated by $[\ \ ]_{\text{seq}}$ is equal to ${\mathcal {T}},$ that is, if ${\mathcal {T}}_{\text{seq}}={\mathcal {T}}.$

References

↑ Eduard Čech, Zdeněk Frolík, Miroslav Katětov, Topological spaces Prague: Academia, Publishing House of the Czechoslovak Academy of Sciences, 1966, Theorem 14 A.9 .
↑ S. Dolecki, An Initiation into Convergence Theory, in F. Mynard, E. Pearl (editors), Beyond Topology, AMS, Contemporary Mathematics, 2009.

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.

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[1] Eduard Čech, Zdeněk Frolík, Miroslav Katětov, Topological spaces Prague: Academia, Publishing House of the Czechoslovak Academy of Sciences, 1966, Theorem 14 A.9 .

[2] S. Dolecki, An Initiation into Convergence Theory, in F. Mynard, E. Pearl (editors), Beyond Topology, AMS, Contemporary Mathematics, 2009.

[1]

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