Countably compact groups and finest totally bounded topologies

Comfort, W. W.; Saks, Victor

doi:10.2140/pjm.1973.49.33

Cited by 50 publications

(13 citation statements)

References 19 publications

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“…(g) In fact, given an abelian group G, the topology induced on G by Hom(G, T) is a totally bounded group topology [33] (1.5) in which every subgroup is closed [31] (2.1).…”

Section: Dense Subgroups: Scattered Resultsmentioning

confidence: 99%

“…As noted [33] (1.6, 1.7) above, for abelian groups G this is the topology induced on G by Hom(G, T). It is shown in [31] (2.2) that when G is infinite abelian, that topology on G is never pseudocompact; that is obvious now, in view of the result cited above in 4(l).…”

Section: Concerning the Group Topologies Suptb(g) And Supps(g T )mentioning

confidence: 88%

“…To the authors' knowledge, this question was first addressed in [31] (2.4), where a straightforward abelian counterexample is offered. Later this fact was noted (see [62] …”

Section: Closed Subgroups Of Pseudocompact Groupsmentioning

confidence: 99%

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Non-Abelian Pseudocompact Groups

Comfort

Remus

2016

Axioms

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“…(g) In fact, given an abelian group G, the topology induced on G by Hom(G, T) is a totally bounded group topology [33] (1.5) in which every subgroup is closed [31] (2.1).…”

Section: Dense Subgroups: Scattered Resultsmentioning

confidence: 99%

Section: Concerning the Group Topologies Suptb(g) And Supps(g T )mentioning

confidence: 88%

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Non-Abelian Pseudocompact Groups

Comfort

Remus

2016

Axioms

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“…For an Abelian group G (without any topology) let G # denote the group G equipped with the maximal totally bounded (Hausdorff) group topology (see [3] or [5] for the definition of this topology and the proof of its existence).…”

Section: Lemma 22 Assume That X Is a Noncompact Metric Space And Y Imentioning

confidence: 99%

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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“…The following statement is available, implicitly or directly, in [6], [9] or [7]. We record it here in passing because it provides some qualitative information about the partially ordered set 98{G)\ our principal interest, however, is quantitative (see 5.3).…”

Section: On the Partially Ordered Set 98(g}mentioning

confidence: 99%