Some results on separate and joint continuity

Let f : X × K → R be a separately continuous function and C a countable collection of subsets of K. Following a result of Calbrix and Troallic, there is a residual set of points x ∈ X such that f is jointly continuous at each point of { x } × Q , where Q is the set of y ∈ K for which the collection C includes a basis of neighborhoods in K. The particular case when the factor K is second countable was recently extended by Moors and Kenderov to any Čech-complete Lindelöf space K and Lindelöf α-favorable X, improving a generalization of Namiokaʹs theorem obtained by Talagrand. Moors proved the same result when K is a Lindelöf p-space and X is conditionally σ-α-favorable space. Here we add new results of this sort when the factor X is σ C ( X ) -β-defavorable and when the assumption “base of neighborhoods” in Calbrix–Troallicʹs result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces.