Convergent sequences in minimal groups
| dc.creator | Shakhmatov, Dmitri | |
| dc.date | 2009-01-01 | |
| dc.date.accessioned | 2026-07-07T12:23:44Z | |
| dc.date.available | 2026-07-07T12:23:44Z | |
| dc.description | A Hausdorff topological group G is minimal if every continuous isomorphism f : G --> H between G and a Hausdorff topological group H is open. Clearly, every compact Hausdorff group is minimal. It is well known that every infinite compact Hausdorff group contains a non-trivial convergent sequence. We extend this result to minimal abelian groups by proving that every infinite minimal abelian group contains a non-trivial convergent sequence. Furthermore, we show that "abelian" is essential and cannot be dropped. Indeed, for every uncountable regular cardinal kappa we construct a Hausdorff group topology T_kappa on the free group F(kappa) with kappa many generators having the following properties: (i) (F(kappa), T_kappa) is a minimal group; (ii) every subset of F(kappa) of size less than kappa is T_kappa-discrete (and thus also T_kappa-closed); (iii) there are no non-trivial proper T_kappa-closed normal subgroups of F(kappa). In particular, all compact subsets of (F(kappa), T_kappa) are finite, and every Hausdorff quotient group of (F(kappa), T_kappa) is minimal (that is, (F(kappa), T_kappa) is totally minimal). | |
| dc.identifier | https://arxiv.org/abs/0901.0175 | |
| dc.identifier | http://arxiv.org/abs/0901.0175 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/214096 | |
| dc.subject | General Topology | |
| dc.subject | Group Theory | |
| dc.subject | 22A05 (Primary); 22C05, 54A10, 54A20, 54A25, 54D25, 54H11 (Secondary) | |
| dc.title | Convergent sequences in minimal groups | |
| dc.type | text |