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

See also

• Eduard Čech

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.