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 ${\displaystyle X}$  is a map ${\displaystyle [\ \ ]_{p}}$ 

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

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

The preclosure operator has to satisfy the following properties:

1. ${\displaystyle [\varnothing ]_{p}=\varnothing \!}$  (Preservation of nullary unions);
2. ${\displaystyle A\subseteq [A]_{p}}$  (Extensivity);
3. ${\displaystyle [A\cup B]_{p}=[A]_{p}\cup [B]_{p}}$  (Preservation of binary unions).

The last axiom implies the following:

4. ${\displaystyle A\subseteq B}$  implies ${\displaystyle [A]_{p}\subseteq [B]_{p}}$ .

Topology

A set ${\displaystyle A}$  is closed (with respect to the preclosure) if ${\displaystyle [A]_{p}=A}$ . A set ${\displaystyle U\subset X}$  is open (with respect to the preclosure) if its complement ${\displaystyle A=X\setminus U}$  is closed. The collection of all open sets generated by the preclosure operator is a topology;^[1] however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.^[2]

Examples

Premetrics

Given ${\displaystyle d}$  a premetric on ${\displaystyle X}$ , then

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

is a preclosure on ${\displaystyle X.}$ 

Sequential spaces

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

See also

• Eduard Čech

References

1. Eduard Čech, Zdeněk Frolík, Miroslav Katětov, Topological spaces Prague: Academia, Publishing House of the Czechoslovak Academy of Sciences, 1966, Theorem 14 A.9 [1].
2. S. Dolecki, An Initiation into Convergence Theory, in F. Mynard, E. Pearl (editors), Beyond Topology, AMS, Contemporary Mathematics, 2009.

• 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.