# Article

 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 . Category: math . 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 . Date available: 2009-09-24T21:22:15Z Last updated: 2015-09-15 Stable URL: http://hdl.handle.net/10338.dmlcz/126189 . Reference: [1] H. Blumberg: New properties of all real functions.Trans. Amer. Math. Soc. 24 (1922), 113-128. MR 1501216, 10.1090/S0002-9947-1922-1501216-9 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 Reference: [3] J. B. Brown: Variations on Blumberg's Theorem.Real Anal. Exchange 9 (1983), 123-137. 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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