Minimal topological groups

Pseudocompact group topologies and totally dense subgroups

“…For G compact, the subgroups H oί G with this latter, weaker property are exactly the subgroups of G with no coarser (Hausdorff) topological group topology. That is: A totally bounded group <^g> admits no coarser group topology if and only if H Π K 3 {e} for every nondegenerate, closed subgroup K of the Weil completion H. These "minimal groups" have been studied extensively (see for example [27], [lo], [30], [15], [16]). The totally dense subgroups of compact groups are exactly those subgroups which, together with all their Hausdorff quotients, are minimal groups [31], [IS].…”

Examples of locally compact noncompact minimal topological groups

“…(X, S) being minimal, one obtains S£ = @, whence (X, 2) is a precompact and complete topological group hence compact (cf. also [4]). It is clear that the above statement remains true, if we only assume (X, X) to be complete in its two-sided uniformity instead of being locally compact.…”

“…In 1971, R. M. Stephenson, Jr., [4], showed that an abelian locally compact topological group must be compact if it is minimal (i.e., if it does not admit a strictly coarser Hausdorff group topology). He left open the question, whether there exist locally compact noncompact minimal topological groups.…”

Topological groups which satisfy an open mapping theorem

“…Since Stephenson [14, Theorem 2] has shown that totally dense subgroups of compact groups are minimal, it also follows that U n is a minimal topological group. Since the product of a minimal group with a compact group is minimal [14], if further follows that U n x K is a minimal topological group, and so a B r (sd) group, for any compact group K.…”