Norm continuity of pointwise quasi-continuous mappings

Mirmostafaee, Alireza Kamel

Mathematica Bohemica, Tome 143 (2018) no. 3, pp. 329-335.

Résumé

Let $X$ be a Baire space, $Y$ be a compact Hausdorff space and $\varphi \colon X \to C_p(Y )$ 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 $X$.

Détail

DOI : 10.21136/MB.2018.0016-17

Classification : 54C05, 54C08, 54C35
Keywords: function space; weak continuity; generalized continuity; quasi-continuous function; pointwise topology