# zbMATH — the first resource for mathematics

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
Zbl 0582.54008
Full Text:
##### References:
 [1] Angosto, C.; Cascales, B.; Namioka, I., Distances to spaces of Baire one functions, Math. Z. 263 (2009), 103-124 · Zbl 1173.54007 [2] Bouziad, A., L’espace de Helly a la propriété de Namioka, C. R. Acad. Sci., Paris, Sér. I 317 (1993), 841-843 French · Zbl 0798.54039 [3] Bouziad, A., Every Čech-analytic Baire semitopological group is a topological group, Proc. Am. Math. Soc. 124 (1996), 953-959 · Zbl 0857.22001 [4] Choquet, G., Lectures on Analysis. Vol. 1: Integration and Topological Vector Spaces, Mathematics Lecture Note Series. W. A. Benjamin Inc., New-York (1969) · Zbl 0181.39601 [5] Christensen, J. P. R., Joint continuity of separately continuous functions, Proc. Am. Math. Soc. 82 (1981), 455-461 · Zbl 0472.54007 [6] Debs, G., Pointwise and uniform convergence on a Corson compact space, Topology Appl. 23 (1986), 299-303 · Zbl 0613.54007 [7] Deville, R., Point convergence and uniform convergence on a compact space, Bull. Pol. Acad. Sci., Math. 37 (1989), 507-515 French · Zbl 0759.54002 [8] Deville, R.; Godefroy, G., Some applications of projective resolutions of identity, Proc. Lond. Math. Soc., III. Ser. 67 (1993), 183-199 · Zbl 0798.46008 [9] Hansel, G.; Troallic, J.-P., Quasicontinuity and Namioka’s theorem, Topology Appl. 46 (1992), 135-149 · Zbl 0812.54019 [10] Haydon, R., Baire trees, bad norms and the Namioka property, Mathematika 42 (1995), 30-42 · Zbl 0870.46008 [11] Kenderov, P. S.; Kortezov, I. S.; Moors, W. B., Norm continuity of weakly continuous mappings into Banach spaces, Topology Appl. 153 (2006), 2745-2759 · Zbl 1100.54014 [12] Mirmostafaee, A. K., Norm continuity of quasi-continuous mappings into {$$C_p(X)$$} and product spaces, Topology Appl. 157 (2010), 530-535 · Zbl 1190.54010 [13] Mirmostafaee, A. K., Quasi-continuity of horizontally quasi-continuous functions, Real Anal. Exch. 39 (2013-2014), 335-344 · Zbl 1322.54009 [14] Mirmostafaee, A. K., Continuity of separately continuous mappings, Math. Slovaca 64 (2014), 1019-1026 · Zbl 1349.54034 [15] Namioka, I., Separate continuity and joint continuity, Pac. J. Math. 51 (1974), 515-531 · Zbl 0294.54010 [16] Oxtoby, J. C., Measure and Category. A Survey of the Analogies between Topological and Measure Spaces, Graduate Texts in Mathematics 2. Springer, New York (1971) · Zbl 0217.09201 [17] Talagrand, M., Espaces de Baire et espaces de Namioka, Math. Ann. 270 (1985), 159-164 French · Zbl 0582.54008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.