𝐹_{𝜎}-set covers of analytic spaces and first class selectors

Hansell, R. W.

doi:10.1090/S0002-9939-1986-0818473-1

$F_ \sigma$-set covers of analytic spaces and first class selectors
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Proc. Amer. Math. Soc. 96 (1986), 365-371 Request permission

Abstract:

Let $X$ be an analytic space (e.g., a complete metric space). We prove that any point-countable ${F_\sigma }$-set cover of $X$ either has $\sigma$-discrete refinement, or else there is a compact subset of $X$ not covered by any countable subfamily of the cover. It follows that any point-countable ${F_\sigma }$-additive family in $X$ has a $\sigma$-discrete refinement. This is used to show that any weakly ${F_\sigma }$-measurable multimap, defined on $X$ and taking nonempty, closed and separable values in a complete metric space, has a selector of the first Baire class.

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