Some results on CDH spaces. I
type: Item
instance of: scholarly article
title: Some results on CDH spaces. I
zbMATH Open document ID: 0647.54017
DOI: 10.1016/0166-8641(88)90006-5
author: S. H. Smith
published in: Topology and its Applications
publication date: 1988-01-01T00:00:00Z
review text: ``A separable space X is said to be a CDH (countable dense homogeneous) space, if for any two countable dense subsets A and B in X, there is an autohomeomorphism h of X such that \(h(A)=B.''\) Manifolds which have a countable base for their topology (countable weight) are CDH (\textit{Bennet} 1972 and \textit{Bessaga, Pelczynski} in 1975). Also the countable powers of both the unit interval, I, and the real line R are known to be CDH (\textit{Fort} 1962). For manifolds the authors construct a separable non-CDH manifold of weight c. The construction is similar to the Rudin-Zenor construction of a perfectly normal non-metrizable manifold. Next they establish that any separable manifold of weight at most b is CDH where b is the minimum cardinality of an unbounded (mod finite) family of functions from \(\omega\) to \(\omega\). They define \[ h(h_ 1)[h_ 2]=\min \{\kappa: 2^{\kappa}(I^{\kappa})[R^{\kappa}]\quad is\quad not\quad CDH\} \] and \(q=\min \{\kappa: 2^{\kappa}\) is not sequentially \(separable\}=\min \{\kappa: there\) is no Q-set of cardinality \(\kappa\}\). Also recall that \(p=\min \{\kappa: MA_{\kappa}\) (\(\sigma\)-centered) fails\(\}\). In the final two sections they establish that \(p\leq h\), \(h_ 1,h_ 2\leq q\). Therefore it follows that, e.g., R c is not CDH and it is independent (of \(\omega_ 1<c)\) that \(R^{\omega_ 1}\) is CDH. Editor's remark: One proof in this paper seems to contain a gap. In [\textit{R. Hernández-Gutiérrez}, Topol. Proc. 56, 125--146 (2020; Zbl 1442.54015)] the author gives a corrected proof.
Mathematics Subject Classification ID: 54C99
Mathematics Subject Classification ID: 54D65
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 4056538
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: Martin's axiom
zbMATH Keywords: CDH (countable dense homogeneous) space
zbMATH Keywords: separable non-CDH manifold of weight c
reviewed by: Alan Dow
MaRDI profile type: Publication
full work available at URL: 0166-8641(88)90006-5
OpenAlex ID: W2131281251
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description: scientific article; zbMATH DE number 4056538