Density character of subgroups of topological groups

Abstract:

We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an $\omega$-narrow topological group $G$: (i) $G$ is homeomorphic to a subspace of a separable regular space; (ii) $G$ is topologically isomorphic to a subgroup of a separable topological group; (iii) $G$ is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group.

A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality $\mathfrak {c}$ of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group $G$ which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that $G$ is homeomorphic to a subspace of a separable Tychonoff space.

We show that every precompact (abelian) topological group of weight less than or equal to $\mathfrak {c}$ is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight $\mathfrak {c}$. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.