Norm continuity of pointwise quasi-continuous mappings

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 $X$.

Keywords

• function space
• weak continuity
• generalized continuity
• quasi-continuous function
• pointwise topology