Algebraic structure of pseudocompact groups

Dikranjan, Dikran; Shakhmatov, Dmitri

doi:10.1090/memo/0633

Cited by 44 publications

(63 citation statements)

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“…Proof. By [7,Lemma 5.3], the variety G is precompact. Hence, the equivalence of items (i)-(iv) follows from Corollary 3.8.…”

Section: Proof the Implication (Iv)⇒(iii) Is Trivial While The Implmentioning

confidence: 99%

“…One can deduce this theorem from [7,Theorem 6.2] but we decided to give a separate proof of it. Another description of the algebraic structure of pseudocompact torsion Abelian groups can be found in [3,Theorem 3.19] and [7,Theorem 6.2].…”

Section: The Algebraic Structure Of Pseudocompact Torsion Abelian Groupsmentioning

confidence: 99%

“…By Lemma 2.3, there exist a dense V-independent selectively sequentially pseudocompact subset Y of H σ and a set C ⊆ σ such that |Y | = |C| = |σ \ C| = σ and Y ∩ H C = {0}. By [7,Lemmas 3.4 and 4.3], there exist a V-independent subset Z of H C and a set…”

mentioning

confidence: 99%

“…Selectively sequentially pseudocompact topologies on V-free groups Definition 3.1. [7] An infinite cardinal τ is called admissible if there exists a pseudocompact group of cardinality τ . Definition 3.2.…”

mentioning

confidence: 99%

“…Since X is dense in H σ , it is also dense in G. Since X is selectively sequentially pseudocompact, Proposition 1.6 implies that G is selectively sequentially pseudocompact as well. Note that the completion of G coincides with H σ , so the rest of the statements follows from [7,Fact 2.10]. Non-precompact varieties are not easy to come by, as many of the known varieties are precompact.…”

mentioning

confidence: 99%

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Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups

Dorantes-Aldama

Shakhmatov

2017

Topology and its Applications

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Abstract. A space X is selectively sequentially pseudocompact if for every family {Un : n ∈ N} of non-empty open subsets of X, one can choose a point xn ∈ Un for every n ∈ N in such a way that the sequence {xn : n ∈ N} has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topology. Since selectively sequentially pseudocompact spaces are strongly pseudocompact in the sense of García-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of García-Ferreira and Tomita.The symbol N denotes the set of natural numbers and N + = N \ {0} denotes the set of positive integer numbers. The symbol Z denotes the group of integer numbers, R the set of real numbers and T the circle group {e iθ : θ ∈ R} ⊆ R 2 . The symbols ω, ω 1 and c stand for the first infinite cardinal, the first uncountable cardinal and the cardinality of the continuum, respectively. For a set X, the set of all finite subsets of X is denoted by [X] <ω , while [X] ω denotes the set of all countably infinite subsets of X.If X is a subset of a group G, then X is the smallest subgroup of G that contains X. For groups which are not necessary Abelian we use the multiplication notation, while for Abelian groups we always use the additive one. In particular, e denotes the identity element of a group and 0 is used for the zero element of an Abelian group. Recall that an element g of a group G is torsion if g n = e for some positive integer n. A group is torsion if all of its elements are torsion. A group G is bounded torsion if there exists n ∈ N such that g n = e for all g ∈ G.Let G be an Abelian group. The symbol t(G) stands for the subgroup of torsion elements of G. For each n ∈ N, note that nG = {ng : g ∈ G} is a subgroup of G and the map g → ng (g ∈ G) is a homomorphism of G onto nG. For a cardinal τ we denote by G (τ ) the direct sum of τ copies of the group G.We will say that a sequence {x n : n ∈ N} of points in a topological space X:• converges to a point x ∈ X if the set {n ∈ N : x n ∈ W } is finite for every open neighbourhood W of x in X; • is convergent if it converges to some point of X.

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“…Proof. By [7,Lemma 5.3], the variety G is precompact. Hence, the equivalence of items (i)-(iv) follows from Corollary 3.8.…”

Section: Proof the Implication (Iv)⇒(iii) Is Trivial While The Implmentioning

confidence: 99%