Quotients of locally minimal groups

A topological group G is called locally minimal if there exists a neighbourhood V of 1 in G such that if H is a Hausdorff group and f : G → H is a continuous isomorphism such that f(V) is a neighbourhood of 1 in H, then f is open. This paper is focused on the study of quotients of locally minimal groups. A topological group G is called locally q-minimal if there exists a neighbourhood V of the identity of G such that whenever H is a Hausdorff group and f : G → H is a continuous surjective homomorphism such that f(V) is a neighbourhood of 1 in H, then f is open. We find a close connection between locally q-minimality and divisibility, by showing that a dense subgroup of Rn is locally q-minimal if and only if it is divisible. A description of the locally q-minimal subgroups of the n-dimensional torus Tn is also given. Two weaker versions of local q-minimality are proposed – a topological group G is: (a) locally t-minimal, if all Hausdorff quotients of G are locally minimal; (b) locally q∗-minimal, if there exists a neighbourhood V of 1 in G such that whenever H is a Hausdorff group and f : G → H is a continuous surjective homomorphism such that f (V ) is a neighbourhood of 1 in H and ker f ⊆ V , then f is open. While local q-minimality and local t-minimality are preserved by taking quotients, local q∗-minimality fails to have this property (so does not coincide with local t-minimality), yet it has a relevant advantage. By means of an appropriate no- tion of local t-density, one can show that a dense subgroup H of a Hausdorff group G is locally q∗-minimal if and only if G is locally q∗-minimal and H is locally t-dense in G. Similar criterion for local q-minimality is not available (examples are given to show that a topological group with a dense locally q-minimal subgroup need not be locally q-minimal). The interrelations of these three versions of local minimality, as well as their prominence properties are studied in detail.