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On Borel-measurable collections of countable unions of finite-dimensional compacta. (English) Zbl 0562.54055

Summary: We consider the following open problem: if B is a Borel set in the product \(X\times Y\) of compact metrizable spaces whose vertical sections B(x) are countable-dimensional (i.e. countable unions of zero-dimensional sets) \(G_{\delta}\)-sets in Y, does there exist a countable-dimensional compactum Z containing topologically all sections B(x)? We show that if each B(x) is covered by countably many compact finite-dimensional sets, the answer is positive and if, in addition, the sections B(x) are compact, one can find a desired compactum Z which is also a countable union of finite-dimensional compacta. These results are based on a recent theorem of D. Cenzer and R. D. Mauldin [Adv. Math. 38, 55-90 (1980; Zbl 0466.03018)] about Borel uniformization of coanalytic sets with nonmeager sections and on some theorems due to C. Dellacherie [in ”Analytic sets”, London (1980; Zbl 0451.04001)] about ”analytic derivatives”, respectively.

MSC:

54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54E45 Compact (locally compact) metric spaces
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
03E15 Descriptive set theory
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