Geodesic
Ideal Banach category theorems and functions
Mathematica Bohemica, Tome 122 (1997) no. 1, pp. 13-20

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

DOI MR   Zbl

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.
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.
DOI : 10.21136/MB.1997.126189
Classification : 54A25, 54B15, 54C08, 54E52
Keywords: Banach Category Theorem; categorical almost continuity; Blumberg space; separate and joint continuity 
Piotrowski, Zbigniew. Ideal Banach category theorems and functions. Mathematica Bohemica, Tome 122 (1997) no. 1, pp. 13-20. doi: 10.21136/MB.1997.126189
@article{10_21136_MB_1997_126189,
     author = {Piotrowski, Zbigniew},
     title = {Ideal {Banach} category theorems and functions},
     journal = {Mathematica Bohemica},
     pages = {13--20},
     year = {1997},
     volume = {122},
     number = {1},
     doi = {10.21136/MB.1997.126189},
     mrnumber = {1446396},
     zbl = {0888.54035},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.1997.126189/}
}
TY  - JOUR
AU  - Piotrowski, Zbigniew
TI  - Ideal Banach category theorems and functions
JO  - Mathematica Bohemica
PY  - 1997
SP  - 13
EP  - 20
VL  - 122
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.1997.126189/
DO  - 10.21136/MB.1997.126189
LA  - en
ID  - 10_21136_MB_1997_126189
ER  - 
%0 Journal Article
%A Piotrowski, Zbigniew
%T Ideal Banach category theorems and functions
%J Mathematica Bohemica
%D 1997
%P 13-20
%V 122
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.1997.126189/
%R 10.21136/MB.1997.126189
%G en
%F 10_21136_MB_1997_126189

[1] H. Blumberg: New properties of all real functions. Trans. Amer. Math. Soc. 24 (1922), 113-128. | DOI | MR

[2] J. C. Bradford C. Goffman: Metric spaces in which Blumberg's theorem holds. Proc. Amer. Math. Soc. 11 (1960), 667-670. | MR

[3] J. B. Brown: Variations on Blumberg's Theorem. Real Anal. Exchange 9 (1983), 123-137.

[4] R. Engelking: General Topology. Warszawa (1977). | MR | Zbl

[5] J. Kaniewski Z. Piotrowski: Concerning continuity apart from a meager set. Proc. Amer. Math. Soc. 98 (1986), 324-328. | DOI | MR

[6] J. Kaniewski Z. Piotrowski D. A. Rose: Ideal Banach category theorems. Rocky Mountain J. Math., (accepted). | MR

[7] P. S. Kenderov: Multi-valued mappings and properties of them similar to continuity. Russian Math. Surveys 35 (1980), 246-249. | DOI

[8] Z. Piotrowski: Blumberg property versus almost continuity. Internat. J. Math. & Math. Sci. 10 (1987), 93-96. | DOI | MR | Zbl

[9] I. Reclaw: Restrictions to continuous functions and Boolean algebras. Proc. Amer. Math. Soc. 118 (1993), 791-796. | DOI | MR | Zbl

[10] B. S. Thomson: Real Functions. Lecture Notes in Mathematics 1170, Springer Verlag, 1985. | MR | Zbl

[11] H. E. White, Jr.: Topological spaces in which Blumberg's theorem holds. Proc. Amer. Math. Soc., 44 (1974), 454-462. | MR | Zbl

Cité par Sources :