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• W2951935670 abstract "We show that all sufficiently nice $lambda$-sets are countable dense homogeneous ($mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $kappa le mathfrak{b}$ there is a countable dense homogeneous metric space of size $kappa$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $kappa$ is equivalent to the existence of a $lambda$-set of size $kappa$. On the other hand, it is consistent with the continuum arbitrarily large that every $mathsf{CDH}$ metric space has size either $omega_1$ or size $mathfrak c$. An example of a Baire $mathsf{CDH}$ metric space which is not completely metrizable is presented. Finally, answering a question of Arhangel'skii and van Mill we show that that there is a compact non-metrizable $mathsf{CDH}$ space in ZFC." @default.
• W2951935670 created "2019-06-27" @default.
• W2951935670 creator A5013610605 @default.
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• W2951935670 creator A5065080063 @default.
• W2951935670 date "2018-09-18" @default.
• W2951935670 modified "2023-09-27" @default.
• W2951935670 title "Countable dense homogeneity and $lambda$-sets" @default.
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• W2951935670 hasPublicationYear "2018" @default.
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