| Title:
|
Ideal Banach category theorems and functions (English) |
| Author:
|
Piotrowski, Zbigniew |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| 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 |
| DOI:
|
10.21136/MB.1997.126189 |
| . |
| Date available:
|
2009-09-24T21:22:15Z |
| Last updated:
|
2020-07-29 |
| 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 |
| . |
