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Zbl 1230.54015
Mirmostafaee, Alireza Kamel
Topological games and strong quasi-continuity. (English)
[J] Banach J. Math. Anal. 5, No. 2, 131-137, electronic only (2011). ISSN 1735-8787/e

For topological spaces $X$, $Y$ and $Z$, a function $ \phi :X \to Z$ is quasi-continuous at a point $x \in X$ if for arbitrary neighborhoods $V$ of $x$ and $W$ of $\phi(x)$, there is an open set $G$ of $V$ such that $\phi (V) \subset W$. The function $ \phi :X \to Z$ is quasi-continuous if it is quasi-continuous at each point of $X$. A function $f: X \times Y \to Z $ is called Kempisty continuous if it is quasi-continuous in the first variable and continuous in the second. A function $f: X \times Y \to Z $ is strongly quasi-continuous at a point $(x,y) \in X \times Y$ if for each neighborhood $W$ of $f(x,y)$ and for each product of open sets $U \times V \in X \times Y$ containing $(x,y)$, there is a nonempty open set $U_1 \in U$ and a neighborhood $V_1 \in V$ of $y$ such that $ f(U_1 \times V_1) \in W$. In the paper under review, the author, using topological games arguments, proves that if $X$ is a Baire space, $Y$ is a $W$-space (a countability property defined by Gruenhage) and $Z$ is a regular space, then every Kempisty continuous function $f: X \times Y \to Z $ is strongly quasi-continuous. He gives an application of his result, in particular, if $Z$ is a Moore space, $X$ is Baire and $Y$ is a Corson space, then every Kempisty continuous function from $X \times Y$ to $Z$ is jointly continuous on a dense subset of $X \times Y$. The author also gives another application of his result to the continuity of group actions.
[Abderrahmane Bouchair (Jijel)]

MSC 2000:: *54C05 Continuous maps
54C30 Real-valued functions on topological spaces
91A44 Games involving topology or set theory
54C35 Function spaces (general topology)
46E15 Banach spaces of functions defined by smoothness properties

Keywords: quasi-continuous mapping; quasi-continuous with respect to one variable; strongly quasi-continuous; topological games

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