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 $[\quad ]_{p}$

[\quad ]_{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:

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 $A=X\setminus U$ 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 $[A]_{p}\subseteq \operatorname {cl}(A)$ for all $A\subset X$ .

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 $[\quad ]_{\text{seq}}$ 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 $[\quad ]_{\text{seq}}$ is equal to ${\mathcal {T}}$ , that is, if ${\mathcal {T}}_{\text{seq}}={\mathcal {T}}$ .

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.

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