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Non-meager \(P\)-filters are countable dense homogeneous. (English) Zbl 1272.54020

A (separable metrizable) space is countable dense homogeneous if any two countable dense sets are homeomorphic by an autohomeomorphism of the ambient space. The authors prove that if \(\mathcal{F}\) is a non-meager \(P\)-filter then, when viewed as a subspace of the Cantor cube \(\{0,1\}^\mathbb{N}\), both it and its infinite power are countable dense homogeneous. This sheds light on Problem 387 of [J. van Mill (ed.) and G. M. Reed (ed.), Open problems in topology. Amsterdam etc.: North-Holland. (1990; Zbl 0718.54001)], which asks for which (zero-dimensional) subsets of the real line the infinite power is countable dense homogeneous. It is known that such spaces must be Baire spaces and if Borel also completely metrizable. The present example adds to a list of consistent examples that are not Borel.
Reviewer: K. P. Hart (Delft)

MSC:

54C99 Maps and general types of topological spaces defined by maps
54A35 Consistency and independence results in general topology
54B10 Product spaces in general topology
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)

Citations:

Zbl 0718.54001
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