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Title: Ideal Banach category theorems and functions (English)
Author: Piotrowski, Zbigniew
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 122
Issue: 1
Year: 1997
Pages: 13-20
Summary lang: English
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Category: math
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Summary: Based on some earlier findings on Banach Category Theorem for some "nice" $\sigma$-ideals by J. Kaniewski, D. Rose and myself I introduce the $h$ operator ($h$ stands for "heavy points") to refine and generalize kernel constructions of A. H. Stone. Having obtained in this way a generalized Kuratowski's decomposition theorem I prove some characterizations of the domains of functions having "many" points of $h$-continuity. Results of this type lead, in the case of the $\sigma$-ideal of meager sets, to important statements of Abstract Analysis such as Blumberg or Namioka-type theorems. (English)
Keyword: Banach Category Theorem
Keyword: categorical almost continuity
Keyword: Blumberg space
Keyword: separate and joint continuity
MSC: 54A25
MSC: 54B15
MSC: 54C08
MSC: 54E52
idZBL: Zbl 0888.54035
idMR: MR1446396
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Date available: 2009-09-24T21:22:15Z
Last updated: 2015-09-15
Stable URL: http://hdl.handle.net/10338.dmlcz/126189
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Reference: [2] J. C. Bradford C. Goffman: Metric spaces in which Blumberg's theorem holds.Proc. Amer. Math. Soc. 11 (1960), 667-670. MR 0146310
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Reference: [4] R. Engelking: General Topology.Warszawa (1977). Zbl 0373.54002, MR 0500780
Reference: [5] J. Kaniewski Z. Piotrowski: Concerning continuity apart from a meager set.Proc. Amer. Math. Soc. 98 (1986), 324-328. MR 0854041, 10.1090/S0002-9939-1986-0854041-3
Reference: [6] J. Kaniewski Z. Piotrowski D. A. Rose: Ideal Banach category theorems.Rocky Mountain J. Math., (accepted). MR 1639861
Reference: [7] P. S. Kenderov: Multi-valued mappings and properties of them similar to continuity.Russian Math. Surveys 35 (1980), 246-249. 10.1070/RM1980v035n03ABEH001845
Reference: [8] Z. Piotrowski: Blumberg property versus almost continuity.Internat. J. Math. & Math. Sci. 10 (1987), 93-96. Zbl 0625.54014, MR 0875967, 10.1155/S0161171287000127
Reference: [9] I. Reclaw: Restrictions to continuous functions and Boolean algebras.Proc. Amer. Math. Soc. 118 (1993), 791-796. Zbl 0781.26003, MR 1152289, 10.1090/S0002-9939-1993-1152289-8
Reference: [10] B. S. Thomson: Real Functions.Lecture Notes in Mathematics 1170, Springer Verlag, 1985. Zbl 0581.26001, MR 0818744
Reference: [11] H. E. White, Jr.: Topological spaces in which Blumberg's theorem holds.Proc. Amer. Math. Soc., 44 (1974), 454-462. Zbl 0295.54017, MR 0341379
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