# 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:

1. $[\varnothing ]_{p}=\varnothing \!$ (Preservation of nullary unions);
2. $A\subseteq [A]_{p}$ (Extensivity);
3. $[A\cup B]_{p}=[A]_{p}\cup [B]_{p}$ (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 $A=X\setminus U$ is closed. The collection of all open sets generated by the preclosure operator is a pretopology.

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