Arkady Leiderman: On subgroups of separable topological groups

Set Theory and Topology seminar (BGU)

On Wednesday, November 19, 16:45 – 18:30, Room -101 of the Mathematics Department.

Speaker: Arkady Leiderman (BGU)

Title: On subgroups of separable topological groups.

Abstract: All spaces and topological groups are assumed to be Hausdorff.
It is well-known that a subspace S of a separable metrizable space X is separable,
but a closed subspace S of a separable Hausdorff topological space X is not necessarily separable. Moreover, a closed linear subspace S of a separable Hausdorff topological vector space X can fail to be separable.

In several classes of topological groups, the situation improves notably. It is known that a closed subgroup S of a separable locally compact topological group G is separable and that a metrizable subgroup of a separable topological group is separable.

In our work we look at conditions on the topological group G which are sufficient to guarantee its separability if G is a subgroup of a separable Hausdorff group X.

We obtained positive results
1) for a large class of pro-Lie groups.
Recall that a topological group is called a pro-Lie group if it is a projective limit of finite-dimensional Lie groups.

2) every feathered subgroup of a separable group is separable.
A topological group G is called feathered if it contains a compact subgroup K such that the quotient space G/K is metrizable.

Some new results in the negative direction are the following:
1) any precompact topological group of weight ≤ continuum is topologically isomorphic to a closed subgroup of a separable pseudocompact group of weight ≤ continuum.
2) Under the Continuum Hypothesis, we present an example of a separable countably compact abelian group G which contains a non-separable closed subgroup. We do not know if such an example exists in ZFC .

This is a joint work with Sidney A. Morris (Australia) and Michael Tkachenko (Mexico).

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