Mathematics > General Topology
[Submitted on 11 Sep 2009]
Title:Some results on separate and joint continuity
View PDFAbstract: 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 Lindelöf space $K$ and Lindelöf $\alpha$-favorable $X$, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when $K$ is a Lindelöf $p$-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.
Submission history
From: Ahmed Bouziad [view email] [via CCSD proxy][v1] Fri, 11 Sep 2009 18:26:42 UTC (16 KB)
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