On non-injectivity of Borel functions selecting points from compact sets (Q1961705)

Let \(X\) be a compact metric space without isolated points and \({\mathcal K}(X)\) the hyperspace consisting of all nonempty closed sets of \(X\) (as the points) and endowed with the Hausdorff distance. It is proved, that each Borel function \(f:{\mathcal K}(X)\to X\) such that \(f(K)\in K\) has the property that \(f^{-1}(x) \cap {\mathcal B}\) is uncountable for any non-meager Borel set \({\mathcal B}\) in \({\mathcal K}(X)\). Therefore, the measure-theoretical selection marriage theorem [\textit{R. D. Mauldin}, Am. J. Math. 104, 823-828 (1982; Zbl 0499.28008)] has no analogue in the Baire category setting.