2012/07/14 by Zbigniew Piotrowski, Piotrowski, Zbigniew, Russell Waller +1 · 1 citation
Mathematics · #54B10 (Secondary) #54C05 (Primary) 54C35 #FOS: Mathematics #General Topology (math.GN) #math.GN #msc:54B10 #msc:54C05 #msc:54C35
paper · pdf · doi:10.48550/arxiv.1207.3436
11 pages
arxiv created 2012/07/14 · arxiv updated 2012/07/17
Recall that a Hausdorff space X is said to be Namioka if for every compact (Hausdorff) space Y and every metric space Z, every separately continuous function f:X×Y→Z is continuous on D×Y for some dense Gδ subset D of X. It is well known that in the class of all metrizable spaces, Namioka and Baire spaces coincide (Saint-Raymond, 1983). Further it is known that every completely regular Namioka space is Baire and that every separable Baire space is Namioka (Saint-Raymond, 1983). In our paper we study spaces X, we call them weakly Namioka, for which the conclusion of the theorem for Namioka spaces holds provided that the assumption of compactness of Y is replaced by second countability of Y. We will prove that in the class of all completely regular separable spaces and in the class of all perfectly normal spaces, X is Baire if and only if it is weakly Namioka.