A topological group observation on the Banach-Mazur separable quotient problem
- Authors: Gabriyelyan, Saak , Morris, Sidney
- Date: 2019
- Type: Text , Journal article
- Relation: Topology and Its Applications Vol. 259, no. (2019), p. 283-286
- Full Text:
- Reviewed:
- Description: The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, T
Density character of subgroups of topological groups
- Authors: Leiderman, Arkady , Morris, Sidney , Tkachenko, Mikhail
- Date: 2017
- Type: Text , Journal article
- Relation: Transactions of the American Mathematical Society Vol. 369, no. 8 (2017), p. 5645-5664
- Full Text:
- Reviewed:
- Description: We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an omega-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.
An open mapping theorem
- Authors: Gabriyelyan, Saak , Morris, Sidney
- Date: 2016
- Type: Text , Journal article
- Relation: Bulletin of the Australian Mathematical Society Vol. 94, no. 1 (2016), p. 65-69
- Full Text:
- Reviewed:
- Description: It is proved that any surjective morphism f : Z(k) -> K onto a locally compact group K is open for every cardinal k. This answers a question posed by Hofmann and the second author.
Nonmeasurable subgroups of compact groups
- Authors: Hernández, Salvador , Hofmann, Karl , Morris, Sidney
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Group Theory Vol. 19, no. 1 (2016), p. 179-189
- Full Text:
- Reviewed:
- Description: In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? An affirmative answer is given for all compact groups with the exception of some metric profinite groups which are almost perfect and strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup. © 2016 by De Gruyter 2016 Generalitat Valenciana PROMETEO/2014/062 We are grateful for our referee's useful comments. In particular, the suggestion that originally we had overlooked [Pacific J. Math. 116 (1985), 217-241] shortened the proof of Theorem 4.3 considerably.
Iwasawa's local splitting theorem for pro-Lie groups
- Authors: Hofmann, Karl , Morris, Sidney
- Date: 2008
- Type: Text , Journal article
- Relation: Forum Mathematicum Vol. 20, no. 4 (2008), p. 607-629
- Full Text:
- Reviewed:
- Description: If the nilradical () of the Lie algebra of a pro-Lie group G is finite dimensional modulo the center (), then every identity neighborhood U of G contains a closed normal subgroup N such that G/N is a Lie group and G and N × G/N are locally isomorphic. © Walter de Gruyter 2008.
- Description: C1
Open mapping theorem for topological groups
- Authors: Hofmann, Karl , Morris, Sidney
- Date: 2007
- Type: Text , Journal article
- Relation: Topology Proceedings Vol. 31, no. 2 (2007), p. 533-551
- Full Text:
- Reviewed:
- Description: We survey sufficient conditions that force a surjective continuous homomorphism between topological groups to be open. We present the shortest proof yet of an open mapping theorem between projective limits of finite dimensional Lie groups.
- Description: C1
- Description: 2003005915