Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0676.54005
Juhász, István
A weakening of $\clubsuit$, with applications to topology. (English)
[J] Commentat. Math. Univ. Carol. 29, No.4, 767-773 (1988). ISSN 0010-2628

With $L\sb 1$ the set of countable limit ordinals, the author denotes by (t) the following combinatorial principle: there are an $\omega\sb 1$-sequence $\{S\sb{\lambda}:$ $\lambda \in L\sb 1\}$, with each $S\sb{\lambda}$ a cofinal $\omega$-sequence in $\lambda$, and for each $\lambda$ a disjoint partition $S\sb{\lambda}=\cup\sb{n<\omega}S\sb{\lambda}(n)$, such that if $X\subseteq \omega\sb 1$ with $\vert X\vert =\omega\sb 1$ then for some $\lambda \in L\sb 1$ (depending on X) we have $\vert X\cap S\sb{\lambda}(n)\vert =\omega$ for all n. That $\clubsuit$ implies (t) is clear; that (t) is consistently weaker than $\clubsuit$ is given by Theorem 1.2: If a single Cohen real is added to a model of ZFC, then (t) holds in the resulting extension. \par The author shows (Theorem 2.1) that (t) implies the existence of an Ostaszewski topology for $\omega\sb 1$-that is, a locally compact Hausdorff topology ${\cal J}$ in which each initial segment in $\omega\sb 1$ is ${\cal J}$-open and each ${\cal J}$-open set is either countable or co-countable; from (t)$+CH$ one may arrange further that $<\omega\sb 1,{\cal J}>$ is countably compact. In a much more complicated inverse limit construction, the author shows that (t) implies the existence of a compact Hausdorff space X such that $t(X)=\pi(X)=\omega$, but $\chi (X)=\omega\sb 1$ for each $p\in X$; such a space X cannot be constructed in ZFC, since {\it A. Dow} has shown [Topology Proc. 13, No.1, 17-72 (1988)] from PFA that if X is compact and $t(X)=\omega$, then $\chi (p,X)=\omega$ for some $p\in X$.
[W.W.Comfort]

MSC 2000:: *54A35 Consistency and independence results (general topology)
54A25 Cardinality properties of topological spaces
54D30 Compactness

Keywords: cardinal functions; Ostaszewski space; tightness; combinatorial principle; Cohen real; model of ZFC; (t)+CH

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]