| Title:
|
Extraresolvability and cardinal arithmetic (English) |
| Author:
|
Alas, O. T. |
| Author:
|
García-Ferreira, S. |
| Author:
|
Tomita, A. H. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
279-292 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Following Malykhin, we say that a space $X$ is {\it extraresolvable\/} if $X$ contains a family $\Cal D$ of dense subsets such that $|\Cal D| > \Delta(X)$ and the intersection of every two elements of $\Cal D$ is nowhere dense, where $\Delta(X) = \min \{|U|: U$ is a nonempty open subset of $X\}$ is the {\it dispersion character\/} of $X$. We show that, for every cardinal $\kappa$, there is a compact extraresolvable space of size and dispersion character $2^\kappa$. In connection with some cardinal inequalities, we prove the equivalence of the following statements: \newline 1) $2^\kappa < 2^{{\kappa}^{+}}$, 2) $(\kappa^{+})^{\kappa}$ is extraresolvable and 3) $A(\kappa^{+})^{\kappa}$ is extraresolvable, where $A(\kappa^{+})$ is the one-point compactification of the discrete space $\kappa^{+}$. For a regular cardinal $\kappa \geq \omega$, we show that the following are equivalent: 1) $2^{< \kappa} < 2^{\kappa}$; 2) $G(\kappa,\kappa)$ is extraresolvable; 3) $G(\kappa,\kappa)^\lambda$ is extraresolvable for all $\lambda < \kappa$; and 4) there exists a space $X$ such that $X^\lambda$ is extraresolvable, for all $\lambda < \kappa$, and $X^\kappa$ is not extraresolvable, where $G(\kappa,\kappa)= \{x \in \{0,1\}^\kappa : |\{ \xi < \kappa : x_\xi \neq 0 \}| < \kappa \}$ for every $\kappa \geq \omega$. It is also shown that if $X$ is extraresolvable and $\Delta(X) = |X|$, then all powers of $X$ have a dense extraresolvable subset, and $\lambda^\kappa$ contains a dense extraresolvable subspace for every cardinal $\lambda \geq 2$ and for every infinite cardinal $\kappa$. For an infinite cardinal $\kappa$, if $2^\kappa > {\frak c}$, then there is a totally bounded, connected, extraresolvable, topological Abelian group of size and dispersion character equal to $\kappa$, and if $\kappa = \kappa^\omega$, then there is an $\omega$-bounded, normal, connected, extraresolvable, topological Abelian group of size and dispersion character equal to $\kappa$. (English) |
| Keyword:
|
extraresolvable |
| Keyword:
|
$\kappa$-resolvable |
| MSC:
|
03E35 |
| MSC:
|
54A25 |
| MSC:
|
54A35 |
| MSC:
|
54F99 |
| idZBL:
|
Zbl 0976.54004 |
| idMR:
|
MR1732649 |
| . |
| Date available:
|
2009-01-08T18:52:08Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119084 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| . |
