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Keywords:
Banach Category Theorem; categorical almost continuity; Blumberg space; separate and joint continuity
Banach Category Theorem; categorical almost continuity; Blumberg space; separate and joint continuity
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.
References:
[1] H. Blumberg: New properties of all real functions. Trans. Amer. Math. Soc. 24 (1922), 113-128. DOI 10.1090/S0002-9947-1922-1501216-9 | MR 1501216
[2] J. C. Bradford C. Goffman: Metric spaces in which Blumberg's theorem holds. Proc. Amer. Math. Soc. 11 (1960), 667-670. MR 0146310
[3] J. B. Brown: Variations on Blumberg's Theorem. Real Anal. Exchange 9 (1983), 123-137.
[4] R. Engelking: General Topology. Warszawa (1977). MR 0500780 | Zbl 0373.54002
[5] J. Kaniewski Z. Piotrowski: Concerning continuity apart from a meager set. Proc. Amer. Math. Soc. 98 (1986), 324-328. DOI 10.1090/S0002-9939-1986-0854041-3 | MR 0854041
[6] J. Kaniewski Z. Piotrowski D. A. Rose: Ideal Banach category theorems. Rocky Mountain J. Math., (accepted). MR 1639861
[7] P. S. Kenderov: Multi-valued mappings and properties of them similar to continuity. Russian Math. Surveys 35 (1980), 246-249. DOI 10.1070/RM1980v035n03ABEH001845
[8] Z. Piotrowski: Blumberg property versus almost continuity. Internat. J. Math. & Math. Sci. 10 (1987), 93-96. DOI 10.1155/S0161171287000127 | MR 0875967 | Zbl 0625.54014
[9] I. Reclaw: Restrictions to continuous functions and Boolean algebras. Proc. Amer. Math. Soc. 118 (1993), 791-796. DOI 10.1090/S0002-9939-1993-1152289-8 | MR 1152289 | Zbl 0781.26003
[10] B. S. Thomson: Real Functions. Lecture Notes in Mathematics 1170, Springer Verlag, 1985. MR 0818744 | Zbl 0581.26001
[11] H. E. White, Jr.: Topological spaces in which Blumberg's theorem holds. Proc. Amer. Math. Soc., 44 (1974), 454-462. MR 0341379 | Zbl 0295.54017
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