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Register a new userSelection properties of the split interval and the Continuum Hypothesis
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Taras Banakh
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We prove that every usco multimap $Phi:Xto Y$ from a metrizable separable space $X$ to a GO-space $Y$ has an $F_sigma$-measurable selection. On the other hand, for the split interval $ddot{mathbb I}$ and the projection $P:ddot{mathbb I}^2to{mathbb I}^2$ of its square onto the unit square ${mathbb I}^2$, the usco multimap $P^{-1}:{mathbb I}^2multimapddot{mathbb I}^2$ has a Borel ($F_sigma$-measurable) selection if and only if the Continuum Hypothesis holds. This CH-example shows that know results on Borel selections of usco maps into fragmentable compact spaces cannot be extended to a wider class of compact spaces.
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A function $f:Xto Y$ between topological spaces is called $sigma$-$continuous$ (resp. $barsigma$-$continuous$) if there exists a (closed) cover ${X_n}_{ninomega}$ of $X$ such that for every $ninomega$ the restriction $f{restriction}X_n$ is continuous. By $mathfrak c_sigma$ (resp. $mathfrak c_{barsigma}$) we denote the largest cardinal $kappalemathfrak c$ such that every function $f:Xtomathbb R$ defined on a subset $Xsubsetmathbb R$ of cardinality $|X|<kappa$ is $sigma$-continuous (resp. $barsigma$-continuous). It is clear that $omega_1lemathfrak c_{barsigma}lemathfrak c_sigmalemathfrak c$. We prove that $mathfrak plemathfrak q_0=mathfrak c_{barsigma}=min{mathfrak c_sigma,mathfrak b,mathfrak q}lemathfrak c_sigmalemin{mathrm{non}(mathcal M),mathrm{non}(mathcal N)}$. The equality $mathfrak c_{barsigma}=mathfrak q_0$ resolves a problem from the initial version of the paper.
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