We present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hernández-Gutiérrez and Hrušák. The method of the proof also allows us to obtain for any n ∈ ω ∪ {∞} an n-dimensional metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.
It was proved in [HM] that each topological group (G,·,τ) may be embedded into a connected topological group (Ĝ,•,τ̂). In fact, two methods of introducing τ̂ were given. In this note we show relations between them.
In this paper we give a complete isomorphical classification of free topological groups of locally compact zero-dimensional separable metric spaces . From this classification we obtain for locally compact zero-dimensional separable metric spaces and that the free topological groups and are isomorphic if and only if and are linearly homeomorphic.
An important consequence of a result of Katětov and Morita states that every metrizable space is contained in a complete metrizable space of the same dimension. We give an equivariant version of this fact in the case of a locally compact -compact acting group.
Every reasonably sized matrix group has an injective homomorphism into the group of all bijections of the natural numbers. However, not every reasonably sized simple group has an injective homomorphism into .
The main results concern commutativity of Hewitt-Nachbin realcompactification or Dieudonné completion with products of topological groups. It is shown that for every topological group that is not Dieudonné complete one can find a Dieudonné complete group such that the Dieudonné completion of is not a topological group containing as a subgroup. Using Korovin’s construction of -dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological...