On the cardinality of countable dense homogeneous spaces. (English) Zbl 1315.54004
A separable topological space \(X\) is countable dense homogeneous (CDH) if for every pair \((D,E)\) of countable dense subsets of \(X\) there exists a homeomorphism \(h:X\longrightarrow X\) such that \(h[D]=E\). An interesting (and uncommon) feature of the paper under review is that it deals with arbitrary Hausdorff CDH spaces, without assuming metrizability.
Theorem 2.1 shows that CDH spaces have size at most continuum. Corollary 2.5 shows that, under the assumption \(2^\omega < 2^{\omega_1}\), compact CDH spaces are first-countable. This result is not provable in ZFC, since J. Steprāns and H. X. Zhou [Topology Appl. 28, No. 2, 147–154 (1988; Zbl 0647.54017)] have shown that \(2^{\omega_1}\) is CDH under \(\mathrm{MA}+\neg\mathrm{CH}\).
Question 1.1 asks whether there exists a ZFC example of a non-metrizable compact CDH space. A natural candidate would be the classical double arrow space \(\mathbb{A}\) of Alexandroff and Urysohn. However, Theorem 3.2 shows that \(\mathbb{A}\) is not CDH. Subsequently, R. Hernández-Gutiérrez [Topology Appl. 160, No. 10, 1123–1128 (2013; Zbl 1284.54023)] proved that \(\mathbb{A}\times 2^\omega\) is not CDH either, thus answering Question 3.3.
Finally, using ideas of K. Kunen [Topology Appl. 12, 283–287 (1981; Zbl 0466.54015)], the authors show that Question 1.1 has an affirmative answer under CH (their example is also hereditarily Lindelöf and hereditarily separable). Subsequently, R. Hernández-Gutiérrez et al. [Fundam. Math. 226, No. 2, 157–172 (2014; Zbl 1301.54050)] showed that Question 1.1 has an affirmative answer in ZFC.
Theorem 2.1 shows that CDH spaces have size at most continuum. Corollary 2.5 shows that, under the assumption \(2^\omega < 2^{\omega_1}\), compact CDH spaces are first-countable. This result is not provable in ZFC, since J. Steprāns and H. X. Zhou [Topology Appl. 28, No. 2, 147–154 (1988; Zbl 0647.54017)] have shown that \(2^{\omega_1}\) is CDH under \(\mathrm{MA}+\neg\mathrm{CH}\).
Question 1.1 asks whether there exists a ZFC example of a non-metrizable compact CDH space. A natural candidate would be the classical double arrow space \(\mathbb{A}\) of Alexandroff and Urysohn. However, Theorem 3.2 shows that \(\mathbb{A}\) is not CDH. Subsequently, R. Hernández-Gutiérrez [Topology Appl. 160, No. 10, 1123–1128 (2013; Zbl 1284.54023)] proved that \(\mathbb{A}\times 2^\omega\) is not CDH either, thus answering Question 3.3.
Finally, using ideas of K. Kunen [Topology Appl. 12, 283–287 (1981; Zbl 0466.54015)], the authors show that Question 1.1 has an affirmative answer under CH (their example is also hereditarily Lindelöf and hereditarily separable). Subsequently, R. Hernández-Gutiérrez et al. [Fundam. Math. 226, No. 2, 157–172 (2014; Zbl 1301.54050)] showed that Question 1.1 has an affirmative answer in ZFC.
Reviewer: Andrea Medini (Wien)
MSC:
| 54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |
| 54D65 | Separability of topological spaces |
| 54H11 | Topological groups (topological aspects) |
References:
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