Joint continuity and quasicontinuity of horizontally quasicontinuous mappings (Q2730512)

Let \(X,Y,Z\) be topological spaces. A mapping \(f: X\times Y\to Z\) is called horizontally quasicontinuous if for any \((x_0,y_0)\in X\times Y\), every neighbourhood \(W\ni f(x_0,y_0)\), every neighbourhood \(U\ni x_0\) in \(X\), and every neighbourhood \(V\ni y_0\) in \(Y\), there exist such a point \((x_1,y_1)\in U\times V\) and a neighbourhood \(U_1\ni x_1\) that \(U_1\subset U\) and \(f(U_1\times \{ y_1\})\subseteq W\). NEWLINENEWLINENEWLINEThe authors show that if \(Y\) is second countable, \(Z\) is metrizable, and \(f\) is horizontally quasicontinuous and continuous with respect to the second variable, then the set of points \(x\in X\), such that \(f\) is continuous at every point from \(\{ x\}\times Y\), is residual in \(X\). NEWLINENEWLINENEWLINEThe notion of horizontal quasicontinuity is used also to refine a result by \textit{N. F. G. Martin} [Duke Math. J. 28, 39-43 (1961; Zbl 0100.18506)] on joint quasicontinuity of separately quasicontinuous mappings.