Preclosure operator

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;Eduard Čech, Zdeněk Frolík, Miroslav Katětov, Topological spaces Prague: Academia, Publishing House of the Czechoslovak Academy of Sciences, 1966, Theorem 14 A.9 https://eudml.org/doc/277000. however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.S. Dolecki, An Initiation into Convergence Theory, in F. Mynard, E. Pearl (editors), Beyond Topology, AMS, Contemporary Mathematics, 2009.

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}.

References

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