## CONCERNING THE DUAL GROUP OF A DENSE SUBGROUP (2002)

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### BibTeX

@MISC{Comfort02concerningthe,

author = {W. W. Comfort and S. U. Raczkowski and F. Javier Trigos-arrieta},

title = {CONCERNING THE DUAL GROUP OF A DENSE SUBGROUP},

year = {2002}

}

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### Abstract

Abstract. Throughout this Abstract, G is a topological Abelian group and ̂G is the space of continuous homomorphisms from G into T in the compact-open topology. A dense subgroup D of G determines G if the (necessarily continuous) surjective isomorphism ̂G ։ ̂D given by h ↦ → h|D is a homeomorphism, and G is determined if each dense subgroup of G determines G. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. (1) There are (many) nonmetrizable, noncompact, determined groups. (2) If the dense subgroup Di determines Gi with Gi compact, then ⊕i Di determines Πi Gi. In particular, if each Gi is compact then ⊕i Gi determines Πi Gi. (3) Let G be a locally bounded group and let G + denote G with its Bohr topology. Then G is determined if and only if G + is determined. (4) Let non(N) be the least cardinal κ such that some X ⊆ T of cardinality κ has positive outer measure. No compact G with w(G) ≥ non(N) is determined; thus if non(N) = ℵ1 (in particular if CH holds), an infinite compact group G is determined if and only if w(G) = ω. Question. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w(G) < κ? Is κ = non(N)? κ = ℵ1?

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Citation Context ...ecessarily a subgroup of G, we denote by S + the set S with the topology inherited from G + . When G is discrete, so that G + = G # , we write S # in place of S + when S ⊆ G. Theorem 0.2. (Glicksberg =-=[17]-=-). Let K ⊆ G ∈ LBA. Then K ∈ K(G) if and only if K + ∈ K(G +). Hence if K ∈ K(G), then K and K + are homeomorphic. Theorem 0.3. (Flor [15]. See also Reid [29]). Let G ∈ LBA and let xn → p ∈ b(G) = b(W... |

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Citation Context ...∈ LCA, then G is compact (in fact G = G + = b(G)). Lemma 4.1 allows a more direct proof in the case that G is discrete. Corollary 4.2. Let G be an infinite Abelian group. determine b(Gd). Lemma 4.3. (=-=[11]-=-). Let G be an Abelian group. Then G # does not (a) If A is a point-separating subgroup of Hom(G, T), then (G, TA) is a totally bounded, Hausdorff topological group with (G, TA) = A; (b) for every t... |

8 |
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Citation Context ...urther if non(N ) = ℵ1, then all seven cardinals mx are equal to ℵ1. The condition non(N ) = ℵ1 is clearly consistent with CH, and it has been shown to be consistent as well with ¬CH (see for example =-=[5]-=-, [15] and [20, Example 1, page 568]), so in particular there are models of ZFC + ¬CH in which every compact (Abelian) group G satisfies: G is determined if and only if G is metrizable. (Without appea... |

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Citation Context ...S + when S ⊆ G. Theorem 0.2. (Glicksberg [16]). Let K ⊆ G ∈ LBA. Then K ∈ K(G) if and only if K + ∈ K(G + ). Hence if K ∈ K(G), then K and K + are homeomorphic. Theorem 0.3. (Flor [14]. See also Reid =-=[28]-=-). Let G ∈ LBA and let xn → p ∈ b(G) = b(W (G)) with each xn ∈ G + ⊆ (W (G)) + ⊆ b(G). Then (a) p ∈ (W (G)) + , and (b) not only xn → p in (W (G)) + ⊆ b(G) but also xn → p in W (G). Remark 0.4. Strict... |

5 |
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Citation Context ...ecessarily a subgroup of G, we denote by S + the set S with the topology inherited from G + . When G is discrete, so that G + = G # , we write S # in place of S + when S ⊆ G. Theorem 0.2. (Glicksberg =-=[16]-=-). Let K ⊆ G ∈ LBA. Then K ∈ K(G) if and only if K + ∈ K(G + ). Hence if K ∈ K(G), then K and K + are homeomorphic. Theorem 0.3. (Flor [14]. See also Reid [28]). Let G ∈ LBA and let xn → p ∈ b(G) = b(... |

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Citation Context ...h(xn) → 0 ∈ T}. Let λ be the Haar measure of Hom(G, T). Then S is a λ-measurable subgroup of Hom(G, T), with λ(S) = 0. Theorem 4.6. Let X be a compact Hausdorff space such that |X| < 2 ℵ1 . Then (a) (=-=[17]-=-) X contains a closed, countably infinite subspace; and (b) X contains a nontrivial convergent sequence. Theorem 4.7. Let G be an Abelian group such that |G| < 2 ℵ1 and let A be a dense subgroup of Ho... |

5 |
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Citation Context ... Further if non(N) = ℵ1, then all seven cardinals mx are equal to ℵ1. The condition non(N) = ℵ1 is clearly consistent with CH, and it has been shown to be consistent as well with ¬CH (see for example =-=[5]-=-, [15] and [20, Example 1, page 568]), so in particular there are models of ZFC + ¬CH in which every compact (Abelian) group G satisfies: G is determined if and only if G is metrizable. (Without appea... |

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Citation Context ...oof. We see in Theorem 4.8 below that compact groups of weight ≥ c are nondetermined. Each such group maps by a continuous homomorphism onto either the group T or a group of the form (Z(p)) ω (p ∈ P) =-=[9]-=-, and such groups are determined by Theorem 1.3. � Discussion 3.4. Obviously an LBA group with no proper dense subgroup is vacuously determined. We mention three classes of such groups.s28 W. W. COMFO... |

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Citation Context ...ite S # in place of S + when S ⊆ G. Theorem 0.2. (Glicksberg [16]). Let K ⊆ G ∈ LBA. Then K ∈ K(G) if and only if K + ∈ K(G + ). Hence if K ∈ K(G), then K and K + are homeomorphic. Theorem 0.3. (Flor =-=[14]-=-. See also Reid [28]). Let G ∈ LBA and let xn → p ∈ b(G) = b(W (G)) with each xn ∈ G + ⊆ (W (G)) + ⊆ b(G). Then (a) p ∈ (W (G)) + , and (b) not only xn → p in (W (G)) + ⊆ b(G) but also xn → p in W (G)... |

4 |
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Citation Context ...al number and denote by l 1 κ the space of real κ-sequences x = {xξ : ξ < κ} such that ||x||1 := � ξ<κ |xξ| < ∞. The additive topological group l 1 κ respects compactness (cf. Remus and TrigosArrieta =-=[29]-=-). The group � (l 1 κ) + is not discrete, so the Weil completion W ((l 1 κ) + ) is another example of a compact nondetermined group. Definition. A topological group G is (group) reflective if the eval... |

4 |
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Citation Context ...rite S # in place of S + when S ⊆ G. Theorem 0.2. (Glicksberg [17]). Let K ⊆ G ∈ LBA. Then K ∈ K(G) if and only if K + ∈ K(G +). Hence if K ∈ K(G), then K and K + are homeomorphic. Theorem 0.3. (Flor =-=[15]-=-. See also Reid [29]). Let G ∈ LBA and let xn → p ∈ b(G) = b(W (G)) with each xn ∈ G + ⊆ (W (G))+ ⊆ b(G). Then (a) p ∈ (W (G))+, and (b) not only xn → p in (W (G))+ ⊆ b(G) but also xn → p in W (G). Re... |

4 | Pseudocompactness on groups, General Topology and Applications, Lecture Notes in Pure and - Trigos-Arrieta - 1991 |

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Citation Context ...ying (additive) topological group of a Montel space. Since by definition these are reflexive LCS, Montel groups are reflective as proven by Smith [31]. Example 5.4. Kōmura [23] and Amemiya and Kōmura =-=[1]-=- construct by induction three different noncomplete Montel spaces, the completion of each being a “big product” of copies of R, and one of them being exactly R c . These groups indicate that Theorem 5... |

3 |
Topologies induced by groups of characters, Fund
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(Show Context)
Citation Context ...ompact (in fact G = G + = b(G)). Lemma 4.1 allows a more direct proof in the case that G is discrete. Corollary 4.2. Let G be an infinite Abelian group. Then G # does not determine b(Gd). Lemma 4.3. (=-=[10]-=-). Let G be an Abelian group. (a) If A is a point-separating subgroup of Hom(G, T), then (G, TA) is a totally bounded, Hausdorff topological group with � (G, TA) = A; (b) for every totally bounded Hau... |

3 |
Some examples of linear topological spaces
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(Show Context)
Citation Context ...tel group we mean the underlying (additive) topological group of a Montel space. Since by definition these are reflexive LCS, Montel groups are reflective as proven by Smith [31]. Example 5.4. Kōmura =-=[23]-=- and Amemiya and Kōmura [1] construct by induction three different noncomplete Montel spaces, the completion of each being a “big product” of copies of R, and one of them being exactly R c . These gro... |

3 |
Dense subgroups of locally compact groups
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(Show Context)
Citation Context ... Discrete groups. (ii) Groups of the form G # = (Gd) + . (It is well known [11, 2.1] that every subgroup of such a group is closed.) (iii) LCA groups of the type given by Rajagopalan and Subrahmanian =-=[27]-=-. Specifically, let κ ≥ ω, fix p ∈ P, and topologize the group G := (Z(p ∞ )) κ so that its subgroup H := (Z(p)) κ in its usual compact topology is open-and-closed in G. Theorem 3.5. (a) A determined ... |

3 |
Sapirovskiĭ, Martin’s axiom and topological spaces, Doklady Akad
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Citation Context ...s 2κ = c. In particular under MA +¬CH it follows from Theorem 4.6(b) that every compact Hausdorff space X such that |X| < 2ℵ1 = c contains a nontrivial convergent sequence. Malykhin and ˇ Sapirovskiĭ =-=[26]-=- have achieved a nontrivial extension of this result: Under MA, every compact Hausdorff space X with |X| ≤ c contains a nontrivial convergent sequence. Theorem 6.1 (MA). Let G be a group with |G| ≤ 2ω... |

2 |
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(Show Context)
Citation Context ...2 κ = c. In particular under MA +¬CH it follows from Theorem 4.6(b) that every compact Hausdorff space X such that |X| < 2 ℵ1 = c contains a nontrivial convergent sequence. Malykhin and ˇ Sapirovskiĭ =-=[25]-=- have achieved a nontrivial extension of this result: Under MA, every compact Hausdorff space X with |X| ≤ c contains a nontrivial convergent sequence. Theorem 6.1 (MA). Let G be a group with |G| ≤ 2 ... |

1 |
to the duality theory of abelian topological groups and to the theory of nuclear groups
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(Show Context)
Citation Context ...onmetrizable, noncompact, determined groups. (2) If the dense subgroup Di determines Gi with Gi compact, then ⊕i Di determines Πi Gi. In particular, if each Gi is compact then ⊕i Gi determines Πi Gi. =-=(3)-=- Let G be a locally bounded group and let G + denote G with its Bohr topology. Then G is determined if and only if G + is determined. (4) Let non(N ) be the least cardinal κ such that some X ⊆ T of ca... |

1 |
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(Show Context)
Citation Context ...Prague Topological Symposium (Praha, August, 2001) and at the 2002 Annual Meeting of the American Mathematical Society (San Diego, January, 2002). A full treatment, with proofs, will appear elsewhere =-=[13]-=-. 23s24 W. W. COMFORT, S. U. RACZKOWSKI, AND F. JAVIER TRIGOS-ARRIETA For each space X = (X, T ) we write K(X) := {K ⊆ X : K is T -compact}. All groups considered here, whether or not equipped with a ... |

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1 | Set theory, Studies in Logic and the Foundations of Mathematics, no. 102, North-Holland Publishing Co - Kunen - 1980 |