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.

A preclosure operator on a set X {\displaystyle X} is a map [     ] p {\displaystyle [\ \ ]_{p}}

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

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

The preclosure operator has to satisfy the following properties:

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

The last axiom implies the following:

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

A set A {\displaystyle A} is closed (with respect to the preclosure) if [ A ] p = A {\displaystyle [A]_{p}=A} . A set U ⊂ X {\displaystyle U\subset X} is open (with respect to the preclosure) if its complement A = X ∖ U {\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]

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

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

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

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

• Eduard Čech

• 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.