A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $U_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}$, such that $U_{\alpha }\subset U_{\beta }$ whenever β ≤ α for all $\alpha ,\beta \in {\mathbb{N}}^{\mathbb{N}}$. The class of all metrizable topological groups is a proper subclass of the class $T{G}_{}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G\in T{G}_{}$ is metrizable, and hence G is strictly angelic. We deduce from this result...

Improving the recent result of the author we show that $indH=0$ is equivalent to $dimH=0$ for every subgroup $H$ of a Hausdorff locally compact group $G$.

The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We apply the more general theory of semicovering spaces to obtain analogous subgroup theorems for topological groups: Every open subgroup of a free Graev topological group is a free Graev topological group. An open subgroup of a free Markov topological group is a free Markov topological group if and only if it is disconnected.

We show that if an uncountable regular cardinal τ and τ + 1 embed in a topological group G as closed subspaces then G is not normal. We also prove that an uncountable regular cardinal cannot be embedded in a torsion free Abelian group that is hereditarily normal. These results are corollaries to our main results about ordinals in topological groups. To state the main results, let τ be an uncountable regular cardinal and G a T₁ topological group. We prove, among others, the following statements:...