Countable dense homogeneous filters and the Menger covering property

TL;DR

This paper constructs a specific non-meager filter in ZFC that is not countable dense homogeneous, and also produces a metrizable Baire topological group with strong local homogeneity but lacking countable dense homogeneity.

Contribution

It provides the first ZFC example of a non-meager filter that is not countable dense homogeneous and introduces a metrizable Baire group with similar properties.

Findings

Constructed a non-meager filter not countable dense homogeneous.

Produced a metrizable Baire topological group with strong local homogeneity.

Answered a question posed by Hernández-Gutiérrez and Hrušák.

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 Hru\v{s}\'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.