Preclosure operator explained

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:l{P}(X)\to} l{P}(X)

where

l{P}(X) is the power set of X.

The preclosure operator has to satisfy the following properties:

[\varnothing] _{p} = \varnothing (Preservation of nullary unions);A \subseteq [A] _{p} (Extensivity);[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;^[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

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

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

See also

• Eduard Čech

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.

Notes and References

1. Eduard Čech, Zdeněk Frolík, Miroslav Katětov, Topological spaces Prague: Academia, Publishing House of the Czechoslovak Academy ofSciences, 1966, Theorem 14 A.9 https://eudml.org/doc/277000.
2. S. Dolecki, An Initiation into Convergence Theory, in F. Mynard, E. Pearl (editors), Beyond Topology,AMS, Contemporary Mathematics, 2009.