×

Mappings of Baire spaces into function spaces and Kadeč renorming. (English) Zbl 0794.54036

Summary: Assuming that there exists in the unit interval \([0,1]\) a coanalytic set of continuum cardinality without any perfect subset, we show the existence of a scattered compact Hausdorff space \(K\) with the following properties: (i) For each continuous map \(f\) on a Baire space \(B\) into \((C(K)\), pointwise), the set of points of continuity of the map \(f:B \to (C(K)\), norm) is a dense \(G_ \delta\) subset of \(B\), and (ii) \(C(K)\) does not admit a Kadeč norm that is equivalent to the supremum norm. This answers the question of Deville, Godefroy and Haydon under the set theoretic assumption stated above.

MSC:

54E52 Baire category, Baire spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bouziad, A., Une classe d’espaces co-Namioka, C.R. Acad. Sci. Paris, 310, 779-782 (1990) · Zbl 0706.54024
[2] Baumgartner, J. E.; Taylor, A. D.; Wagon, S., Structural properties of ideals (1982), Warszawa: PWN, Warszawa · Zbl 0549.03036
[3] Debs, G., Poinwise and uniform convergence on a Corson compact space, Topology and its Applications, 23, 299-303 (1986) · Zbl 0613.54007 · doi:10.1016/0166-8641(85)90047-1
[4] R. Deville,Continuité séparée et jointe sur certains produits d’espaces topologiques, These 3e cycle, Paris, 1984.
[5] Deville, R., Convergence ponctuelle et uniforme sur un espace compact, Bull. Acad. Polon. Sci., 37, 7-12 (1989) · Zbl 0759.54002
[6] R. Deville and G. Godefroy,Some applications of projectional resolutions of identity, to appear. · Zbl 0798.46008
[7] Engelking, R., General Topology (1977), Warszawa: PWN, Warszawa · Zbl 0373.54002
[8] Frankiewicz, R.; Kunen, K., Solution of Kuratowski’s problem on function having the Baire property I, Fund. Math., 128, 171-180 (1987) · Zbl 0646.54006
[9] Haydon, R., A counter example to several questions about scattered compact spaces, Bull. London Math. Soc., 22, 261-268 (1990) · Zbl 0725.46007 · doi:10.1112/blms/22.3.261
[10] Haydon, R.; Rogers, C. A., A locally uniformly convex renorming for certain C(K), Mathematika, 37, 1-8 (1990) · Zbl 0725.46008 · doi:10.1112/S0025579300012754
[11] Jech, T., Set Theory (1978), New York-San Francisco-London: Academic Press, New York-San Francisco-London · Zbl 0419.03028
[12] J. E. Jayne, I. Namioka and C. A. Rogers,Topological properties related to the Radon-Nikodým property, to appear.
[13] J. E. Jayne, I. Namioka and C. A. Rogers,σ-fragmentable Banach spaces, to appear. · Zbl 0761.46009
[14] Koppelberg, S., General Theory of Boolean Algebras (1989), Amsterdam-New York-Oxford-Tokyo: North-Holland, Amsterdam-New York-Oxford-Tokyo
[15] Krom, M. R., Certain products of metric Baire spaces, Proc. Am. Math. Soc., 42, 588-593 (1974) · Zbl 0276.54005 · doi:10.2307/2039550
[16] Kuratowski, K., Topology I (1966), Warszawa: PWN, Warszawa
[17] Namioka, I., Separate continuity and joint continuity, Pacific J. Math., 51, 515-531 (1974) · Zbl 0294.54010
[18] Negrepontis, S., Banach Spaces and Topology Handbook of Set Theoretic Topology, 1045-1142 (1984), Amsterdam-New York: North-Holland, Amsterdam-New York · Zbl 0584.46007
[19] Porter, J. R.; Woods, R. G., Extensions and absolute of Hausdorff spaces (1988), New York-Berlin-Heidelberg-London-Paris-Tokyo: Springer-Verlag, New York-Berlin-Heidelberg-London-Paris-Tokyo · Zbl 0652.54016
[20] Rogers, C. A.; Jayne, J. E., K-analytic sets, in Analytic Sets, 1-181 (1980), London-New York: Academic Press, London-New York
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.