  A subspace Y of a separable metrizable space X is separable, but without X
metrizable this is not true even If Y is a closed linear subspace of a
topological vector space X. K.H. Hofmann and S.A. Morris introduced the class
of pro-Lie groups which consists of projective limits of finite-dimensional Lie
groups and proved that it contains all compact groups, locally compact abelian
groups and connected locally compact groups and is closed under products and
closed subgroups. A topological group G is almost connected if the quotient
group of G by the connected component of its identity is compact. We prove that
an almost connected pro-Lie group is separable iff its weight is not greater
than c. It is deduced that an almost connected pro-Lie group is separable if
and only if it is a subspace of a separable Hausdorff space. It is 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. Every precompact topological group of weight less than or
equal to c is topologically isomorphic to a closed subgroup of a separable
pseudocompact group of weight c. This implies that there is a wealth of closed
nonseparable subgroups of separable pseudocompact groups. An example is
presented under CH of a separable countably compact abelian group which
contains a non-separable closed subgroup. It is proved that the following
conditions are equivalent for an omega-narrow topological group G: (i) G is a
subspace of a separable regular space; (ii) G is a subgroup of a separable
topological group; (iii) G is a closed subgroup of a separable pathconnected
locally pathconnected group.
