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• W2922081786 abstract "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 R n is locally q -minimal if and only if it is divisible. A description of the locally q -minimal subgroups of the n -dimensional torus T n 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 notion 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." @default.
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• W2922081786 date "2019-07-01" @default.
• W2922081786 modified "2023-09-28" @default.
• W2922081786 title "Quotients of locally minimal groups" @default.
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• W2922081786 doi "https://doi.org/10.1016/j.jmaa.2019.03.010" @default.
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