Countable dense homogeneous filters and the Menger covering property
Abstract
In this note we present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hern\'andez-Guti\'errez and Hrus\'ak. The method of the proof also allows us to obtain a metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.
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