Countable dense homogeneity and λ - sets
نویسندگان
چکیده
We show that all sufficiently nice λ-sets are countable dense homogeneous (CDH). From this fact we conclude that for every uncountable cardinal κ ≤ b there is a countable dense homogeneous metric space of size κ. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size κ is equivalent to the existence of a λ-set of size κ. On the other hand, it is consistent with the continuum arbitrarily large that every CDH metric space has size either ω1 or c. An example of a Baire 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 CDH space in ZFC.
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