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 $FM\left(X\right)$ of locally compact zero-dimensional separable metric spaces $X$. From this classification we obtain for locally compact zero-dimensional separable metric spaces $X$ and $Y$ that the free topological groups $FM\left(X\right)$ and $FM\left(Y\right)$ are isomorphic if and only if $C_p\left(X\right)$ and $C_p\left(Y\right)$ are linearly homeomorphic.

Using CH we construct examples of sequential topological groups: 1. a pair of countable Fréchet topological groups whose product is sequential but is not Fréchet, 2. a countable Fréchet and ${\alpha }_1$ topological group which contains no copy of the rationals.

We prove some extension theorems involving uniformly continuous maps of the universal Urysohn space. As an application, we prove reconstruction theorems for certain groups of autohomeomorphisms of this space and of its open subsets.

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 $G$ that is not Dieudonné complete one can find a Dieudonné complete group $H$ such that the Dieudonné completion of $G\times H$ is not a topological group containing $G\times H$ as a subgroup. Using Korovin’s construction of $G_{\delta }$-dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological...