| Title:
|
Topological characterization of the small cardinal $i$ (English) |
| Author:
|
Franco-Filho, Antonio de Padua |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
44 |
| Issue:
|
4 |
| Year:
|
2003 |
| Pages:
|
745-750 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that the small cardinal number $i = \min \{\vert \Cal A \vert : \Cal A$ is a maximal independent family\} has the following topological characterization: $i = \min \{\kappa \leq c: \{0,1\}^{\kappa}$ has a dense irresolvable countable subspace\}, where $\{0,1\}^{\kappa}$ denotes the Cantor cube of weight $\kappa$. As a consequence of this result, we have that the Cantor cube of weight $c$ has a dense countable submaximal subspace, if we assume (ZFC plus $i=c$), or if we work in the Bell-Kunen model, where $i = {\aleph_{1}}$ and $c = {\aleph_{\omega_1}}$. (English) |
| Keyword:
|
independent family |
| Keyword:
|
irresolvable |
| Keyword:
|
submaximal |
| MSC:
|
54A05 |
| MSC:
|
54A25 |
| MSC:
|
54A35 |
| MSC:
|
54B05 |
| MSC:
|
54B10 |
| MSC:
|
54C25 |
| idZBL:
|
Zbl 1098.54003 |
| idMR:
|
MR2062891 |
| . |
| Date available:
|
2009-01-08T19:32:43Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119429 |
| . |
| Reference:
|
[ASTTW] Alas O.T., Sanchis M., Tkačenko M.G., Tkachuk V.V., Wilson R.G.: Irresolvable and submaximal spaces: homogeneity vs ${\sigma}$-discreteness and new ZFC examples.Topology Appl. 107 (2000), 259-278. MR 1779814 |
| Reference:
|
[BK] Bell M., Kunen K.: On the Pi-character of ultrafilters.C.R. Math. Rep. Acad. Sci. Canada 3 (1981), 351-356. Zbl 0475.54001, MR 0642449 |
| Reference:
|
[Ma] Malykhin V.I.: Irresolvable countable spaces of weight less than $\frak c$.Comment. Math. Univ. Carolinae 40.1 (1999), 181-185. Zbl 1060.54500, MR 1715211 |
| . |
