First countability, \( \omega \)-well-filtered spaces and reflections. (English) Zbl 1440.54017
The authors investigate two new classes of subsets in \(T_0\) spaces, which they call \(\omega\)-Rudin sets and \(\omega\)-well-filtered determined sets. These lie between the class of all closures of countable directed subsets and that of irreducible closed subsets. They also consider two new types of spaces which they call \(\omega\)-\(d\)-spaces and \(\omega\)-well-filtered spaces.
They show that an \(\omega\)-well-filtered \(T_0\) space is locally compact if and only if it is core compact. A consequence is that every core compact well-filtered space is sober, answering the Jia-Jung problem in a new way.
They also prove that all irreducible closed subsets in a first countable \(\omega\)-well-filtered \(T_0\) space are directed. They conclude that a first countable \(T_0\) space \( X\) is sober if and only if \(X\) is well-filtered if and only if \(X\) is an \(\omega\)-well-filtered \(d\)-space.
Making use of \(\omega\)-well-filtered determined sets, they finally give a construction of the \(\omega\)-well-filtered reflections of \(T_0\) spaces and prove that products of \(\omega\)-well-filtered spaces are \(\omega\)-well-filtered.
They show that an \(\omega\)-well-filtered \(T_0\) space is locally compact if and only if it is core compact. A consequence is that every core compact well-filtered space is sober, answering the Jia-Jung problem in a new way.
They also prove that all irreducible closed subsets in a first countable \(\omega\)-well-filtered \(T_0\) space are directed. They conclude that a first countable \(T_0\) space \( X\) is sober if and only if \(X\) is well-filtered if and only if \(X\) is an \(\omega\)-well-filtered \(d\)-space.
Making use of \(\omega\)-well-filtered determined sets, they finally give a construction of the \(\omega\)-well-filtered reflections of \(T_0\) spaces and prove that products of \(\omega\)-well-filtered spaces are \(\omega\)-well-filtered.
Reviewer: Hans Peter Künzi (Rondebosch)
MSC:
| 54D99 | Fairly general properties of topological spaces |
| 54B30 | Categorical methods in general topology |
| 54B20 | Hyperspaces in general topology |
| 06F30 | Ordered topological structures |
Keywords:
sober space; well-filtered space; \( \omega \)-well-filtered space; \(d\)-space; \( \omega\)-\(d\)-space; first countable spaceReferences:
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