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Henriksen and Woods [Topology Appl. 97 (1999) 175–205, Problem (C), p. 203] asked whether there are Tychonoff spaces X and Y with X × Y being Baire such that: (a) Every separately continuous function f :X×Y→ R has a dense (in fact: G δ ) set C ( f ) of points of continuity; (b) There exists a separately continuous function g :X×Y→ R for which C ( g ) fails to contain either A × Y or X × B for any dense G δ set A ⊂ X or dense G δ set B ⊂ Y . We will answer this question by showing the spaces X and Y can even be complete metric and condition (b) can be strengthened to the following: There exists a separately continuous function g :X×Y→ R so that if C ( g ) contains either A × Y or X × B , then both A and B are empty. |
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