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Spaces of functions with countably many discontinuities. (English) Zbl 1133.54010

Generalizing some results from the literature the authors prove: let \(K\) be a separable and pointwise compact set of functions on a Polish space \(\Gamma\). If each function in \(K\) has only countably many discontinuities, then \(C(K)\) admits a pointwise lower semicontinuous, locally uniformly rotund norm (which is equivalent to the supremum norm). If \(K\) is not separable, then \(C(K)\) with the topology of pointwise convergence is \(\sigma\)-fragmentable by its norm.

MSC:

54C35 Function spaces in general topology
46B99 Normed linear spaces and Banach spaces; Banach lattices
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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References:

[1] S. Arygros, P. Dodos and V. Kanellopoulos, Tree structures associated to a family of functions, Journal of Symbolic Logic 70 (2005), 681–695. · Zbl 1083.03044
[2] J. Bourgain, D. H. Fremlin and M. Talagrand, Pointwise compact sets of Baire-measurable functions, American Journal of Mathematics 100 (1978), 845–886. · Zbl 0413.54016
[3] A. Bouziad, L’espace de Helly à la propiété de Namioka, Comptes Rendus de l’Académie des Sciences, Paris, Série I 317 (1993), 841–843. · Zbl 0798.54039
[4] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renorming in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific & Technical, Harlow, 1993. · Zbl 0782.46019
[5] R. Deville and G. Godefroy, Some applications of projective resolutions of identity, Proceedings of the London Mathematical Society 67 (1993), 183–199. · Zbl 0798.46008
[6] R. G. Haydon, Trees in renorming theory, Proceedings of the London Mathematical Society 78 (1999), 549–584. · Zbl 1036.46003
[7] R. G. Haydon, Locally uniformly rotund norms in Banach spaces and their duals, submitted. · Zbl 1158.46005
[8] J. E. Jayne, I. Namioka and C. A. Rogers, Topological properties of Banach spaces, Proceedings of the London Mathematical Society 66 (1993), 651–672. · Zbl 0793.54026
[9] A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. · Zbl 0819.04002
[10] I. Kortezov, The function space over the Helly compact is sigma-fragmentable, Topology and its Applications 106 (2000), 69–75. · Zbl 0974.46022
[11] A. Moltó, J. Orihuela, S. Troyanski and M. Valdivia, A non-linear transfer technique for renorming, Pre-Publicaciones del Departamento de Matematicas, Universidad de Murcia, 20, 2003. · Zbl 1182.46001
[12] E. Odell and H. P. Rosenthal, A double dual characterization of separable Banach spaces containing 1, Israel Journal of Mathematics 20 (1975), 375–384. · Zbl 0312.46031
[13] H. P. Rosenthal, Point-wise compact subsets of the first Baire class, American Journal of Mathematics 99 (1977), 362–378. · Zbl 0392.54009
[14] S. Todorcevic, Trees and linearly ordered sets, in Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 235–293.
[15] S. Todorcevic, Representing trees as relatively compact subsets of the first Baire class, Bulletin. Classe des Sciences Mathématiques et Naturelles. Sciences Mathématiques 30 (2005), 29–45. · Zbl 1144.46009
[16] S. Todorcevic, Compact subsets of the first Baire class, Journal of the American Mathematical Society 12 (1999), 1179–1212. · Zbl 0938.26004
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