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Borel measures in consonant spaces. (English) Zbl 0852.54009

Summary: A topology \(\mathcal T\) on a set \(X\) is called consonant if the Scott topology of the lattice \(\mathcal T\) is compactly generated; equivalently, if the upper Kuratowski topology and the co-compact topology on closed sets of \(X\) coincide. It is proved that every completely regular consonant space is a Prohorov space, and that every first countable regular consonant space is hereditarily Baire. If \(X\) is metrizable separable and co-analytic, then \(X\) is consonant if and only if \(X\) is Polish. Finally, we prove that every pseudocompact topological group which is consonant is compact. Several problems of Dolecki, Greco and Lechicki, of Nogura and Shakhmatov, are solved.

MSC:

54B20 Hyperspaces in general topology
28A51 Lifting theory
54D50 \(k\)-spaces
22A05 Structure of general topological groups
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