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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.
##### MSC:
 54E52 Baire category, Baire spaces 54C60 Set-valued maps in general topology 28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
##### Keywords:
Baire space; Baire property; first category; Baire category
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