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 [math]\displaystyle{ X }[/math] is a map [math]\displaystyle{ [\ \ ]_p }[/math]
- [math]\displaystyle{ [\ \ ]_p:\mathcal{P}(X) \to \mathcal{P}(X) }[/math]
where [math]\displaystyle{ \mathcal{P}(X) }[/math] is the power set of [math]\displaystyle{ X. }[/math]
The preclosure operator has to satisfy the following properties:
- [math]\displaystyle{ [\varnothing]_p = \varnothing \! }[/math] (Preservation of nullary unions);
- [math]\displaystyle{ A \subseteq [A]_p }[/math] (Extensivity);
- [math]\displaystyle{ [A \cup B]_p = [A]_p \cup [B]_p }[/math] (Preservation of binary unions).
The last axiom implies the following:
- 4. [math]\displaystyle{ A \subseteq B }[/math] implies [math]\displaystyle{ [A]_p \subseteq [B]_p }[/math].
Topology
A set [math]\displaystyle{ A }[/math] is closed (with respect to the preclosure) if [math]\displaystyle{ [A]_p=A }[/math]. A set [math]\displaystyle{ U \subset X }[/math] is open (with respect to the preclosure) if its complement [math]\displaystyle{ A = X \setminus U }[/math] 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 [math]\displaystyle{ d }[/math] a premetric on [math]\displaystyle{ X }[/math], then
- [math]\displaystyle{ [A]_p = \{x \in X : d(x,A)=0\} }[/math]
is a preclosure on [math]\displaystyle{ X. }[/math]
Sequential spaces
The sequential closure operator [math]\displaystyle{ [\ \ ]_\text{seq} }[/math] is a preclosure operator. Given a topology [math]\displaystyle{ \mathcal{T} }[/math] with respect to which the sequential closure operator is defined, the topological space [math]\displaystyle{ (X,\mathcal{T}) }[/math] is a sequential space if and only if the topology [math]\displaystyle{ \mathcal{T}_\text{seq} }[/math] generated by [math]\displaystyle{ [\ \ ]_\text{seq} }[/math] is equal to [math]\displaystyle{ \mathcal{T}, }[/math] that is, if [math]\displaystyle{ \mathcal{T}_\text{seq} = \mathcal{T}. }[/math]
See also
References
- ↑ Eduard Čech, Zdeněk Frolík, Miroslav Katětov, Topological spaces Prague: Academia, Publishing House of the Czechoslovak Academy of Sciences, 1966, Theorem 14 A.9 [1].
- ↑ 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.
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