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Preclosure operator

Closure operator

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Source updated April 25, 2026

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.

01Definition

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

02Topology

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; however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.

03Examples

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 ${\mathcal {T}}_{\text{seq}}={\mathcal {T}}.$

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Sources and credits

This article is adapted from the Wikipedia article “Preclosure operator”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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