Suitable sets for topological groups

Comfort, W. W.; Morris, Sidney A.; Robbie, Desmond A.; Svetlichny, Sergey; Tkačenko, Michael G.

doi:10.1016/s0166-8641(97)00129-6

Cited by 24 publications

(19 citation statements)

References 7 publications

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“…We will topologize G to obtain our desired topological groups. The relation between suitability and countably compact groups without non-trivial convergent sequences was first noticed in [5]. We further explore this relation in finite products.…”

Section: The Examples and Some Preliminariesmentioning

confidence: 81%

“…The group H n does not have a suitable set since it is a countably compact group of order 2 without non-trivial convergent sequences (see [5]). 2 Example 2.2.…”

Section: The Examples and Some Preliminariesmentioning

confidence: 99%

“…Hofmann and Morris showed that every locally compact group has a suitable set; in [5] the study of the nonlocally compact case was started and it was followed in [8,9,22,26,29]. It is easy to see that if a power of H has a suitable set then every larger power also does.…”

Section: Introductionmentioning

confidence: 99%

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Countable compactness and finite powers of topological groups without convergent sequences

Tomita¹

2005

Topology and its Applications

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Section: The Examples and Some Preliminariesmentioning

confidence: 81%

“…The group H n does not have a suitable set since it is a countably compact group of order 2 without non-trivial convergent sequences (see [5]). 2 Example 2.2.…”

Section: The Examples and Some Preliminariesmentioning

confidence: 99%

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Countable compactness and finite powers of topological groups without convergent sequences

Tomita¹

2005

Topology and its Applications

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“…The result also immediately follows from a much stronger fact that every infinite subset of G # contains an infinite closed and relatively discrete subset [9]. 4 The referee has kindly suggested that the following comment should be added to our manuscript: "We used above the fact that G # has only finite compact sets. This is a particular case Our next theorem demonstrates that "metric" cannot be omitted from Corollaries 16 and 20.…”

Section: Lemma 25 For Every Abelian Group G the Group Gmentioning

confidence: 90%

A characterization of compactly generated metric groups

Fujita

Shakhmatov

2002

Proc. Amer. Math. Soc.

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Abstract. Recall that a topological group G is: (a) σ-compact if G = {Kn : n ∈ N} where each Kn is compact, and (b) compactly generated if G is algebraically generated by some compact subset of G. Compactly generated groups are σ-compact, but the converse is not true: every countable nonfinitely generated discrete group (for example, the group of rational numbers or the free (Abelian) group with a countable infinite set of generators) is a counterexample. We prove that a metric group G is compactly generated if and only if G is σ-compact and for every open subgroup H of G there exists a finite set F such that F ∪ H algebraically generates G. As a corollary, we obtain that a σ-compact metric group G is compactly generated provided that one of the following conditions holds: (i) G has no proper open subgroups, (ii) G is dense in some connected group (in particular, if G is connected itself), (iii) G is totally bounded (= subgroup of a compact group). Our second major result states that a countable metric group is compactly generated if and only if it can be generated by a sequence converging to its identity element (eventually constant sequences are not excluded here). Therefore, a countable metric group G can be generated by a (possibly eventually constant) sequence converging to its identity element in each of the cases (i), (ii) and (iii) above. Examples demonstrating that various conditions cannot be omitted or relaxed are constructed. In particular, we exhibit a countable totally bounded group which is not compactly generated.If G is a group and X ⊆ G, then we use X to denote the smallest subgroup of G which contains X. A set X algebraically generates G if G = X . Recall that a topological group G is:n ∈ N} where each K n is compact, and (ii) compactly generated if G = K for some compact subset K of G.Obviously, each compactly generated group is σ-compact. The converse is not true in general: any countable nonfinitely generated discrete group (for example, the group of rational numbers or the free (Abelian) group with a countable infinite set of generators) is a counterexample. However, Pestov [15] established that every σ-compact topological group is topologically isomorphic to a closed subgroup of some compactly generated group.

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“…(b) Every topologically simple group in S belongs to S t . In particular, the symmetric topological groups SX and all metrizable topologically simple groups belong to S t (since the transpositions form a suitable set for SX in the ®rst case and by [3,Theorem 6.6] in the second case).…”

Section: Introductionmentioning

confidence: 99%

Locally compact Abelian groups with totally suitable sets

Boschi¹

2001

Journal of Group Theory

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We study the class S t of topological groups G that admit a totally suitable set, i.e., a relatively discrete subset S t Gnf1g such that S f1g is closed in G and N hSi is dense in N for each closed normal subgroup N of G. We resolve an open question from [10] by giving a complete description of the locally compact Abelian groups in S t . We also give a characterization of the groups of p-adic numbers in terms of totally suitable sets.

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Suitable sets for topological groups

Cited by 24 publications

References 7 publications

Countable compactness and finite powers of topological groups without convergent sequences

Countable compactness and finite powers of topological groups without convergent sequences

A characterization of compactly generated metric groups

Locally compact Abelian groups with totally suitable sets

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