A note on iterated duals of certain topological spaces. (English) Zbl 1030.54006
A set is called \(\tau\)-saturated if it is the intersection of its neighborhoods; the topology \(\tau^d\) generated by the complements of \(\tau\)-saturated \(\tau\)-compact subsets of \(X\) is called the dual of the topology \(\tau\) on \(X\). It was known that for \(T_1\)-spaces the relation \(\tau^{dd}=\tau^{dddd}\) holds and the author proves as the main result that this relation holds not only for \(T_1\)-topologies.
He adds that “after this paper was accepted, Martin Kovář of the Technical University of Brno communicated to the author a proof that \(\tau^{dd}=\tau^{dddd}\) for all topologies”. However the paper contains some other results on iterated duals, too.
He adds that “after this paper was accepted, Martin Kovář of the Technical University of Brno communicated to the author a proof that \(\tau^{dd}=\tau^{dddd}\) for all topologies”. However the paper contains some other results on iterated duals, too.
Reviewer: A.A.Ivanov (St.Peterburg)
MSC:
| 54B17 | Adjunction spaces and similar constructions in general topology |
| 54E55 | Bitopologies |
| 54B20 | Hyperspaces in general topology |
| 54B99 | Basic constructions in general topology |
| 54D30 | Compactness |
| 54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |
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