Open subgroups and Pontryagin duality

Banaszczyk, Wojciech; Chasco, M.J.; Martín-Peinador, Elena

doi:10.1007/bf02571709

Cited by 26 publications

(16 citation statements)

References 4 publications

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“…Let f : G → K be a continuous homomorphism onto K, where K is either T c or Z(n) c . Since the homomorphism f is open, H = f −1 (P ) is a proper dense pseudocompact subgroup of G. The restriction ϕ = f ↾ H is an open continuous homomorphism with a compact kernel N. Since the image P = ϕ(H) ∼ = H/N is a reflexive group, we conclude that so is H (see [6,Theorem 2.6]).…”

Section: Remark 43mentioning

confidence: 84%

“…Then H is a proper subgroup of G and, since the homomorphism f is open, H is dense in G. Let ϕ be the restriction of f to H.Then ϕ is open and ker f = ker ϕ, i.e., the homomorphism ϕ has the compact kernel N. Hence the group H is almost metrizable, while Lemma 3.5 implies that H is Baire. Since N is compact and the quotient group H 0 ∼ = H/N is not reflexive, neither is H (see[6, Theorem 2.6]). Note that H satisfies ORC according to Proposition 2.9.…”

mentioning

confidence: 99%

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Pontryagin duality in the class of precompact Abelian groups and the Baire property

Bruguera

Tkachenko²

2012

Journal of Pure and Applied Algebra

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a b s t r a c tWe present a wide class of reflexive, precompact, non-compact, Abelian topological groups G determined by three requirements. They must have the Baire property, satisfy the open refinement condition, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group G ∧ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group G of weight ≥ 2 ω , we find proper dense subgroups H 1 and H 2 of G such that H 1 is reflexive and pseudocompact, while H 2 is non-reflexive and almost metrizable.

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Section: Remark 43mentioning

confidence: 84%

mentioning

confidence: 99%

Pontryagin duality in the class of precompact Abelian groups and the Baire property

Bruguera

Tkachenko²

2012

Journal of Pure and Applied Algebra

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“…As it was noticed in [18] before Definition 2.33, for every T -sequence u in an infinite Abelian group G the subgroup u is open in (G, τ u ) (see also Lemma 2.2), and hence, by Lemmas 1.4 and 2.2 of [3], the following sequences are exact:…”

mentioning

confidence: 79%

On T-Characterized Subgroups of Compact Abelian Groups

Gabriyelyan

2015

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We say that a subgroup H of an infinite compact Abelian group X is T -characterized if there is a Tsequence u = {un} in the dual group of X such that H = {x ∈ X : (un, x) → 1}. We show that a closed subgroup H of X is T -characterized if and only if H is a G δ -subgroup of X and the annihilator of H admits a Hausdorff minimally almost periodic group topology. All closed subgroups of an infinite compact Abelian group X are T -characterized if and only if X is metrizable and connected. We prove that every compact Abelian group X of infinite exponent has a T -characterized subgroup which is not an Fσ-subgroup of X that gives a negative answer to Problem 3.3 in [10].

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“…Indeed, by definition, u k is a T -sequence and (G, u) = (G, u k ). Since, by definition, u k is open, it is dually closed and dually embedded [6]. By Lemma 4, we have n(G, u) ⊂ u k .…”

Section: Lemma 4 Let H Be a Dually Closed And Dually Embedded Subgromentioning

confidence: 92%

On T-sequences and characterized subgroups

Gabriyelyan

2010

Topology and its Applications

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Let X be a compact metrizable abelian group and u = {u n } be a sequence in its dual group X ∧ . Set s u (X) = {x: (u n , x) → 1} and T H 0 = {(z n ) ∈ T ∞ : z n → 1}. Let G be a subgroup of X. We prove that G = s u (X) for some u iff it can be represented as some dually closed subgroup G u of Cl X G × T H 0 . In particular, s u (X) is polishable. Let u = {u n } be a Tsequence. Denote by ( X, u) the group X ∧ equipped with the finest group topology in which u n → 0. It is proved that ( X, u) ∧ = G u and n( X, u) = s u (X) ⊥ . We also prove that the group generated by a Kronecker set cannot be characterized.We shall write our abelian groups additively. For a topological group X , X denotes the group of all continuous characters on X . We denote its dual group by X ∧ , i.e. the group X endowed with the compact-open topology. A group X equipped with the discrete topology is denoted by X d . Denote by n( X) = χ ∈ X ker χ the von Neumann radical of X . X is named Pontryagin reflexive or reflexive if the canonical homomorphism α X : X → X ∧∧ , x → (χ → (χ , x)) is a topological isomorphism. If H is a subgroup of X , we denote by H ⊥ its annihilator. Let X and Y be topological groups and ϕ : X → Y a continuous homomorphism. We denote by ϕ * : Y ∧ → X ∧ , χ → χ • ϕ, the dual homomorphism of ϕ. Let A be a subset of a group X .

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Open subgroups and Pontryagin duality

Cited by 26 publications

References 4 publications

Pontryagin duality in the class of precompact Abelian groups and the Baire property

Pontryagin duality in the class of precompact Abelian groups and the Baire property

On T-Characterized Subgroups of Compact Abelian Groups

On T-sequences and characterized subgroups

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