Topological games and strong quasi-continuity (Q638179)

For topological spaces \(X\), \(Y\) and \(Z\), a function \( \phi :X \to Z\) is quasi-continuous at a point \(x \in X\) if for arbitrary neighborhoods \(V\) of \(x\) and \(W\) of \(\phi(x)\), there is an open set \(G\) of \(V\) such that \(\phi (V) \subset W\). The function \( \phi :X \to Z\) is quasi-continuous if it is quasi-continuous at each point of \(X\). A function \(f: X \times Y \to Z \) is called Kempisty continuous if it is quasi-continuous in the first variable and continuous in the second. A function \(f: X \times Y \to Z \) is strongly quasi-continuous at a point \((x,y) \in X \times Y\) if for each neighborhood \(W\) of \(f(x,y)\) and for each product of open sets \(U \times V \in X \times Y\) containing \((x,y)\), there is a nonempty open set \(U_1 \in U\) and a neighborhood \(V_1 \in V\) of \(y\) such that \( f(U_1 \times V_1) \in W\). In the paper under review, the author, using topological games arguments, proves that if \(X\) is a Baire space, \(Y\) is a \(W\)-space (a countability property defined by Gruenhage) and \(Z\) is a regular space, then every Kempisty continuous function \(f: X \times Y \to Z \) is strongly quasi-continuous. He gives an application of his result, in particular, if \(Z\) is a Moore space, \(X\) is Baire and \(Y\) is a Corson space, then every Kempisty continuous function from \(X \times Y\) to \(Z\) is jointly continuous on a dense subset of \(X \times Y\). The author also gives another application of his result to the continuity of group actions.