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Norm continuity of pointwise quasi-continuous mappings. (English) Zbl 1474.54064
M. Talagrand in [Math. Ann. 270, 159–164 (1985; Zbl 0582.54008)] provided an example of a pointwise continuous mapping \(\varphi:X\to C_p(X)\), where \(X\) is an \(\alpha\)-favorable space \(X\) which is nowhere norm continuous. The result of Talagrand raises the following question: What are the compact spaces \(Y\) such that for every Baire space \(X\), any continuous (or quasi-continuous) mapping \(\varphi:X\to C_p(Y)\) must be norm continuous at each point of some dense \(G_\delta\) subset of \(X\)? It is known that every \(\alpha\)-favorable space is a Baire space. In this paper the author proves the following result. Let \(X\) be a topological space and let \(\varphi:X\to C_p(Y)\) be a quasi-continuous mapping. Suppose that \(X\) is \(\alpha\)-favorable and \(\mathfrak{b}\) has no winning strategy in \(\mathcal{G}_1(H)\) or \(X\) is Baire and \(\mathfrak{a}\) has a winning strategy in \(\mathcal{G}_1(H)\). If \(Y\) is \(\mathfrak{b}\)-unfavorable for the game \(\mathcal{G}_2(H)\), there is a dense \(G_\delta\) subset \(D\) of \(X\) such that \(\varphi\) is norm continuous on \(D\).
Reviewer: Shou Lin (Ningde)
MSC:
54C35 Function spaces in general topology
54C08 Weak and generalized continuity
54C05 Continuous maps
91A44 Games involving topology, set theory, or logic
Citations:
Zbl 0582.54008
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