Starting with a very simple proof of Frol’ık’s theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order $2$. We also apply Frol’ık’s theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every...

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...