| Title:
|
Products of non-$\sigma $-lower porous sets (English) |
| Author:
|
Rmoutil, Martin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
205-217 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the present article we provide an example of two closed non-$\sigma $-lower porous sets $A, B \subseteq \mathbb R $ such that the product $A\times B$ is lower porous. On the other hand, we prove the following: Let $X$ and $Y$ be topologically complete metric spaces, let $A\subseteq X$ be a non-$\sigma $-lower porous Suslin set and let $B\subseteq Y$ be a non-$\sigma $-porous Suslin set. Then the product $A\times B$ is non-$\sigma $-lower porous. We also provide a brief summary of some basic properties of lower porosity, including a simple characterization of Suslin non-$\sigma $-lower porous sets in topologically complete metric spaces. (English) |
| Keyword:
|
topologically complete metric space |
| Keyword:
|
abstract porosity |
| Keyword:
|
$\sigma $-porous set |
| Keyword:
|
$\sigma $-lower porous set |
| Keyword:
|
Cartesian product |
| MSC:
|
28A05 |
| MSC:
|
54B10 |
| MSC:
|
54E35 |
| MSC:
|
54G20 |
| idZBL:
|
Zbl 1274.28005 |
| idMR:
|
MR3035507 |
| DOI:
|
10.1007/s10587-013-0014-4 |
| . |
| Date available:
|
2013-03-01T16:16:41Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143180 |
| . |
| Reference:
|
[1] Engelking, R.: General Topology. Rev. and Compl. Ed., Sigma Series in Pure Mathematics 6.Heldermann Berlin (1989). MR 1039321 |
| Reference:
|
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| Reference:
|
[3] Zajíček, L.: Porosity and $\sigma$-porosity.Real Anal. Exch. 13 (1987/88), 314-350. Zbl 0666.26003, MR 0943561, 10.2307/44151885 |
| Reference:
|
[4] Zajíček, L.: Smallness of sets of nondifferentiability of convex functions in non-separable Banach spaces.Czech. Math. J. 41 (1991), 288-296. Zbl 0768.58005, MR 1105445 |
| Reference:
|
[5] Zajíček, L.: Products of non-$\sigma$-porous sets and Foran systems.Atti Semin. Mat. Fis. Univ. Modena 44 (1996), 497-505. Zbl 0877.54023, MR 1428780 |
| Reference:
|
[6] Zajíček, L.: On $\sigma$-porous sets in abstract spaces.Abstr. Appl. Anal. 5 (2005), 509-534. MR 2201041, 10.1155/AAA.2005.509 |
| Reference:
|
[7] Zajíček, L., Zelený, M.: Inscribing closed non-$\sigma$-lower porous sets into Suslin non-$\sigma$-lower porous sets.Abstr. Appl. Anal. 3 (2005), 221-227. Zbl 1091.28001, MR 2197116, 10.1155/AAA.2005.221 |
| Reference:
|
[8] Zelený, M., Pelant, J.: The structure of the $\sigma$-ideal of $\sigma$-porous sets.Commentat. Math. Univ. Carol. 45 (2004), 37-72. Zbl 1101.28001, MR 2076859 |
| . |
