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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 $\left[quad\right]_p$

$\left[quad\right]_p:mathcal\left\{P\right\}\left(X\right) to mathcal\left\{P\right\}\left(X\right)$

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

The preclosure operator has to satisfy the following properties:

1. $\left[varnothing\right]_p = varnothing !$ (Preservation of nullary unions);
2. $A subseteq \left[A\right]_p$ (Extensivity);
3. $\left[A cup B\right]_p = \left[A\right]_p cup \left[B\right]_p$ (Preservation of binary unions).

The last axiom implies the following:

4. $A subseteq B$ implies $\left[A\right]_p subseteq \left[B\right]_p$.

Topology

A set $A$ is closed (with respect to the preclosure) if $\left[A\right]_p=A$. A set $Usubset X$ is open (with respect to the preclosure) if $A=Xsetminus U$ 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 $\left[A\right]_psubseteq operatorname\left\{cl\right\}\left(A\right)$ for all $Asubset X$.

Examples

Premetrics

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

$\left[A\right]_p=\left\{xin X : d\left(x,A\right)=0\right\}$

is a preclosure on $X$.

Sequential spaces

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