# Some results on separate and joint continuity

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1 LMRS - Laboratoire de Mathématiques Raphaël Salem

Abstract : 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 \v Cech-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 $\sigma {CX}$-$\beta$-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.

Keywords : separate continuity joint continuity Namioka spaces

Autor: Aicha Bareche - Ahmed Bouziad -

Fuente: https://hal.archives-ouvertes.fr/

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