| Title:
|
The structure of the $\sigma$-ideal of $\sigma$-porous sets (English) |
| Author:
|
Zelený, Miroslav |
| Author:
|
Pelant, Jan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
37-72 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show a general method of construction of non-$\sigma$-porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non-$\sigma$-porous Suslin subset of a topologically complete metric space contains a non-$\sigma$-porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non-$\sigma$-porous element. Namely, if we denote the space of all compact subsets of a compact metric space $E$ with the Vietoris topology by $\Cal K(E)$, then it is shown that each analytic subset of $\Cal K(E)$ containing all countable compact subsets of $E$ contains necessarily an element, which is a non-$\sigma$-porous subset of $E$. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed non-$\sigma$-porous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the $\sigma$-ideal of compact $\sigma$-porous sets. (English) |
| Keyword:
|
$\sigma$-porosity |
| Keyword:
|
descriptive set theory |
| Keyword:
|
$\sigma$-ideal |
| Keyword:
|
trigonometric series |
| Keyword:
|
sets of uniqueness |
| MSC:
|
26E99 |
| MSC:
|
28A05 |
| MSC:
|
42A63 |
| MSC:
|
54H05 |
| idZBL:
|
Zbl 1101.28001 |
| idMR:
|
MR2076859 |
| . |
| Date available:
|
2009-05-05T16:43:15Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119436 |
| . |
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| . |
