Let $X,Y$ be completely regular Baire spaces. Is it true that every real valued separately continuous function on $X\times Y$ has a point of continuity?
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1$\begingroup$ Real valued function? $\endgroup$– Dieter KadelkaCommented Nov 16, 2020 at 12:22
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$\begingroup$ Yes. Thank you for the question. Real or complex valued. I modify the text accordingly. $\endgroup$– Gergo KissCommented Nov 16, 2020 at 15:59
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1$\begingroup$ @DieterKadelka: I think it's worthwhile to point out that if it is true for real-valued functions, then it is also true for all functions with values in completely regular (Hausdorff) spaces (since closed sets and points in such spaces can be separated by continuous real-valued functions). $\endgroup$– Jochen GlueckCommented Nov 16, 2020 at 17:14
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1$\begingroup$ Concering the questions itself: This sounds rather strong to me. There's a famous theorem of Ellis which says that if $(G,\circ)$ is a group, $G$ is endowed with a compact Hausdorff topology and $\circ$ is separately continuous, then $\circ$ is automatically jointly continuous. The assertion in your question would immediately yield Ellis' theorem as a special case (by using the observation from my previous comment). $\endgroup$– Jochen GlueckCommented Nov 16, 2020 at 17:15
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$\begingroup$ You probably know this already, but there are no counterexamples with metrizable $X$ and $Y$ $\endgroup$– Alessandro CodenottiCommented Nov 17, 2020 at 23:25
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A counterexample to this problem (with $X$ Baire and $Y$ compact) was recently constructed by Mykhaylyuk and Pol in this preprint.
answered Nov 18, 2020 at 0:03
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$\begingroup$ Thank you very much. This clearly answers even a stronger question. $\endgroup$ Commented Nov 26, 2020 at 15:07