| Title:
|
Interpolation of $\kappa$-compactness and PCF (English) |
| Author:
|
Juhász, István |
| Author:
|
Szentmiklóssy, Zoltán |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
50 |
| Issue:
|
2 |
| Year:
|
2009 |
| Pages:
|
315-320 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We call a topological space $\kappa$-compact if every subset of size $\kappa$ has a complete accumulation point in it. Let $\Phi(\mu,\kappa,\lambda)$ denote the following statement: $\mu < \kappa < \lambda = \operatorname{cf} (\lambda)$ and there is $\{ S_\xi : \xi < \lambda \} \subset [\kappa]^\mu$ such that $|\{ \xi : |S_\xi \cap A| = \mu \}| < \lambda$ whenever $A \in [\kappa]^{<\kappa}$. We show that if $\Phi(\mu,\kappa,\lambda)$ holds and the space $X$ is both $\mu$-compact and $\lambda$-compact then $X$ is $\kappa$-compact as well. Moreover, from PCF theory we deduce $\Phi(\operatorname{cf} (\kappa), \kappa, \kappa^+)$ for every singular cardinal $\kappa$. As a corollary we get that a linearly Lindelöf and $\aleph_\omega$-compact space is uncountably compact, that is $\kappa$-compact for all uncountable cardinals $\kappa$. (English) |
| Keyword:
|
complete accumulation point |
| Keyword:
|
$\kappa$-compact space |
| Keyword:
|
linearly Lindelöf space |
| Keyword:
|
PCF theory |
| MSC:
|
03E04 |
| MSC:
|
54A25 |
| MSC:
|
54D30 |
| idZBL:
|
Zbl 1212.03029 |
| idMR:
|
MR2537839 |
| . |
| Date available:
|
2009-08-18T12:25:32Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133436 |
| . |
| Reference:
|
[1] Arhangel'skii A.V.: Homogeneity and complete accumulation points.Topology Proc. 32 (2008), 239--243. Zbl 1170.54009, MR 1500085 |
| Reference:
|
[2] Shelah S.: Cardinal Arithmetic.Oxford Logic Guides, vol. 29, Oxford University Press, Oxford, 1994. Zbl 0864.03032, MR 1318912 |
| Reference:
|
[3] van Douwen E.: The Integers and Topology.in Handbook of Set-Theoretic Topology, K. Kunen and J.E. Vaughan, Eds., North-Holland, Amsterdam, 1984, pp. 111--167. Zbl 0561.54004, MR 0776619 |
| . |
