Quasi-continuity of horizontally quasi-continuous functions (Q2356302)

A function \(f:X\times Y\to Z\) of topological spaces is horizontally quasi-continuous if for each \((x_0,y_0)\in X\times Y\), each basic open \(U\times V\subset X\times Y\) with \((x_0,y_0)\in U\times V\) and open \(W\) with \(f(x_0,y_0)\in W\subset Z\) there are open \(U_1\subset U\) and \(y_1\in V\) such that \(f(U_1\times\{y_1\})\subset W\). Theorems of the form ``for suitable \(X\), \(Y\) and \(Z\), if \(f:X\times Y\to Z\) is horizontally quasi-continuous and (quasi-)continuous with respect to the second variable then \(f\) is jointly quasi-continuous'' are proven. Typically \(X\) is Baire, \(Y\) satisfies a countability condition and \(Z\) a regularity condition.