Praclosure operator

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In topology, a praclosure operator or Čech closure operator is a map between subsets of a set, similar to a closure operator, except that it is not required to be idempotent. That is, a praclosure operator obeys only three of the four Kuratowski closure axioms.

1 Definition
2 Topology
3 Examples
- 3.1 Prametrics
- 3.2 Sequential spaces
4 References

[edit] Definition

A praclosure 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 praclosure 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)

[edit] Topology

A set $A$ is closed (with respect to the praclosure) if $[A] p = A$ . A set $U\subset X$ is open (with respect to the praclosure) if $A=X\setminus U$ is closed. The collection of all open sets generated by the praclosure operator is a topology.

The closure operator cl on this topological space satisfies $[A]_p\subseteq \operatorname{cl}(A)$ for all $A\subset X$ .

[edit] Examples

[edit] Prametrics

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

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

is a praclosure on $X$ .

[edit] Sequential spaces

The sequential closure operator $[\quad]_\mbox{seq}$ is a praclosure 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 $\mathcal{T}_\mbox{seq}$ generated by $[\quad]_\mbox{seq}$ is equal to $\mathcal{T}$ , that is, if $\mathcal{T}_\mbox{seq}=\mathcal{T}$ .