Some results on separate and joint continuity

• Published in: arXiv.org
• Main Authors: Bareche, Aicha, Bouziad, Ahmed
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 11.09.2009
• Subjects: Collection; Continuity (mathematics)
• ISSN: 2331-8422

Let \(f: X\times K\to \mathbb R\) be a separately continuous function and \(\mathcal C\) a countable collection of subsets of \(K\). Following a result of Calbrix and Troallic, there is a residual set of points \(x\in X\) such that \(f\) is jointly continuous at each point of \(\{x\}\times Q\), where \(Q\) is the set of \(y\in K\) for which the collection \(\mathcal C\) 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 Lindel\"of space \(K\) and Lindel\"of \(\alpha\)-favorable \(X\), improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when \(K\) is a Lindel\"of \(p\)-space and \(X\) is conditionally \(\sigma\)-\(\alpha\)-favorable space. Here we add new results of this sort when the factor \(X\) is \(\sigma_{C(X)}\)-\(\beta\)-defavorable and when the assumption "base of neighborhoods" in Calbrix-Troallic's result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces.