Transversals in non-discrete groups

Abstract

The concept of `topological right transversal' is introduced to study right transversals in topological groups. Given any right quasigroup $S$ with a Tychonoff topology $T$, it is proved that there exists a Hausdorff topological group in which $S$ can be embedded algebraically and topologically as a right transversal of a subgroup (not necessarily closed). It is also proved that if a topological right transversal $(S,T_S,T^S,\circ )$ is such that $T_S=T^S$ is a locally compact Hausdorff topology on $S$, then $S$ can be embedded as a right transversal of a closed subgroup in a Hausdorff topological group which is universal in some sense.

7 pages