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Concerning a certain \(\sigma\)-algebra in compact Hausdorff spaces. (English) Zbl 0816.54022
This is a continuation of the author’s studies carried on in [Topology Appl. 41, No. 1/2, 73-112 (1991; Zbl 0773.46012)]. Let \(B\) be a Baire space and \(T\) a compact space. The main result (Theorem 1) states that if \(F \subset T\) and (i) \(\{b\in B: \phi(b) \cap F \neq \emptyset\}\) has the Baire property and (ii) \(\{b \in B: \phi(b) \cap F \neq \emptyset\) and \(\phi(b) \cap (T\smallsetminus F) \neq \emptyset\}\) is of the first category for any minimal upper semicontinuous compact valued map \(\phi\) defined on \(B\) with values in \(T\) then \(F\) has the Baire property. Also some other results in the spirit of the Baire category can be found.
54E52 Baire category, Baire spaces
54C60 Set-valued maps in general topology
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
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