Ideal Banach category theorems and functions

Zbigniew Piotrowski

Mathematica Bohemica (1997)

  • Volume: 122, Issue: 1, page 13-20
  • ISSN: 0862-7959

Abstract

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Based on some earlier findings on Banach Category Theorem for some “nice” σ -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 σ -ideal of meager sets, to important statements of Abstract Analysis such as Blumberg or Namioka-type theorems.

How to cite

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Piotrowski, Zbigniew. "Ideal Banach category theorems and functions." Mathematica Bohemica 122.1 (1997): 13-20. <http://eudml.org/doc/248151>.

@article{Piotrowski1997,
abstract = {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.},
author = {Piotrowski, Zbigniew},
journal = {Mathematica Bohemica},
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},
language = {eng},
number = {1},
pages = {13-20},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Ideal Banach category theorems and functions},
url = {http://eudml.org/doc/248151},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Piotrowski, Zbigniew
TI - Ideal Banach category theorems and functions
JO - Mathematica Bohemica
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 122
IS - 1
SP - 13
EP - 20
AB - 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.
LA - eng
KW - Banach Category Theorem; categorical almost continuity; Blumberg space; separate and joint continuity; Banach Category Theorem; categorical almost continuity; Blumberg space; separate and joint continuity
UR - http://eudml.org/doc/248151
ER -

References

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  1. H. Blumberg, 10.1090/S0002-9947-1922-1501216-9, Trans. Amer. Math. Soc. 24 (1922), 113-128. (1922) MR1501216DOI10.1090/S0002-9947-1922-1501216-9
  2. J. C. Bradford C. Goffman, Metric spaces in which Blumberg's theorem holds, Proc. Amer. Math. Soc. 11 (1960), 667-670. (1960) MR0146310
  3. J. B. Brown, Variations on Blumberg's Theorem, Real Anal. Exchange 9 (1983), 123-137. (1983) 
  4. R. Engelking, General Topology, Warszawa (1977). (1977) Zbl0373.54002MR0500780
  5. J. Kaniewski Z. Piotrowski, 10.1090/S0002-9939-1986-0854041-3, Proc. Amer. Math. Soc. 98 (1986), 324-328. (1986) MR0854041DOI10.1090/S0002-9939-1986-0854041-3
  6. J. Kaniewski Z. Piotrowski D. A. Rose, Ideal Banach category theorems, Rocky Mountain J. Math., (accepted). MR1639861
  7. P. S. Kenderov, 10.1070/RM1980v035n03ABEH001845, Russian Math. Surveys 35 (1980), 246-249. (1980) DOI10.1070/RM1980v035n03ABEH001845
  8. Z. Piotrowski, 10.1155/S0161171287000127, Internat. J. Math. & Math. Sci. 10 (1987), 93-96. (1987) Zbl0625.54014MR0875967DOI10.1155/S0161171287000127
  9. I. Reclaw, 10.1090/S0002-9939-1993-1152289-8, Proc. Amer. Math. Soc. 118 (1993), 791-796. (1993) Zbl0781.26003MR1152289DOI10.1090/S0002-9939-1993-1152289-8
  10. B. S. Thomson, Real Functions, Lecture Notes in Mathematics 1170, Springer Verlag, 1985. (1985) Zbl0581.26001MR0818744
  11. H. E. White, Jr., Topological spaces in which Blumberg's theorem holds, Proc. Amer. Math. Soc., 44 (1974), 454-462. (1974) Zbl0295.54017MR0341379

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