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 $[\ \ ]_{p}$

$[\ \ ]_{p}:{\mathcal {P}}(X)\to {\mathcal {P}}(X)$

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

The preclosure operator has to satisfy the following properties:

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 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.

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

See also

• Eduard Čech

References

• A.V. Arkhangelskii, L.S. Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.
• B. Banaschewski, Bourbaki's Fixpoint Lemma reconsidered, Comment. Math. Univ. Carolinae 33 (1992), 303–309.