Mathematica Bohemica, Vol. 143, No. 3, pp. 329-335, 2018
Norm continuity of pointwise quasi-continuous mappings
Alireza Kamel Mirmostafaee
Received February 13, 2017. First published January 17, 2018.
Abstract: Let $X$ be a Baire space, $Y$ be a compact Hausdorff space and $\varphi X \to C_p(Y )$ be a quasi-continuous mapping. For a proximal subset $H$ of $Y \times Y$ we will use topological games $\mathcal{G}_1(H)$ and $\mathcal{G}_2(H)$ on $Y \times Y$ between two players to prove that if the first player has a winning strategy in these games, then $\varphi$ is norm continuous on a dense $G_\delta$ subset of $X$. It follows that if $Y$ is Valdivia compact, each quasi-continuous mapping from a Baire space $X$ to $C_p(Y)$ is norm continuous on a dense $G_\delta$ subset of $X$.
Affiliations: Alireza Kamel Mirmostafaee, Center of Excellence in Analysis on Algebraic Structures, Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Mashhad, Iran, e-mail: mirmostafaei@um.ac.ir
