Extremal phenomena in certain classes of totally bounded groups

W. W. Comfort; Lewis C. Robertson

Displaying similar documents to “Extremal phenomena in certain classes of totally bounded groups”

The dual group of a dense subgroup

William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta (2004)

Czechoslovak Mathematical Journal

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Throughout this abstract, $G$ is a topological Abelian group and $\hat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $𝕋$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\hat{G} ↠ \hat{D}$ given by $h \mapsto h | D$ 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...

Metrization criteria for compact groups in terms of their dense subgroups

Dikran Dikranjan, Dmitri Shakhmatov (2013)

Fundamenta Mathematicae

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According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or $G_{δ}$ -dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined...

Cellularity and the index of narrowness in topological groups

Mihail G. Tkachenko (2011)

Commentationes Mathematicae Universitatis Carolinae

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We study relations between the cellularity and index of narrowness in topological groups and their $G_{δ}$ -modifications. We show, in particular, that the inequalities $in ({(H)}_{τ}) \leq 2^{τ \cdot in (H)}$ and $c ({(H)}_{τ}) \leq 2^{2^{τ \cdot in (H)}}$ hold for every topological group $H$ and every cardinal $τ \geq ω$ , where ${(H)}_{τ}$ denotes the underlying group $H$ endowed with the $G_{τ}$ -modification of the original topology of $H$ and $in (H)$ is the index of narrowness of the group $H$ . Also, we find some bounds for the complexity of continuous real-valued functions $f$ on an arbitrary $ω$ -narrow group...

On a generalization of Abelian sequential groups

Saak S. Gabriyelyan (2013)

Fundamenta Mathematicae

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Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G, τ)}^{\land}$ is a dense -closed subgroup of the compact group ${(G_{d})}^{\land}$ , where $G_{d}$ is the group G with...

Ordinals in topological groups

Raushan Z. Buzyakova (2007)

Fundamenta Mathematicae

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We show that if an uncountable regular cardinal τ and τ + 1 embed in a topological group G as closed subspaces then G is not normal. We also prove that an uncountable regular cardinal cannot be embedded in a torsion free Abelian group that is hereditarily normal. These results are corollaries to our main results about ordinals in topological groups. To state the main results, let τ be an uncountable regular cardinal and G a T₁ topological group. We prove, among others, the following...

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong Gao, Yuanlin Li, Pingping Zhao, Jujuan Zhuang (2016)

Colloquium Mathematicae

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Let G be an additive finite abelian group. For every positive integer ℓ, let $d i s c_{ℓ} (G)$ be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine $d i s c_{ℓ} (G)$ for certain finite groups, including cyclic groups, the groups $G = C ₂ \oplus C_{2 m}$ and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum...

The generalized criterion of Dieudonné for valuated $p$ -groups

Peter Vassilev Danchev (2006)

Acta Mathematica Universitatis Ostraviensis

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We prove that if $G$ is an abelian $p$ -group with a nice subgroup $A$ so that $G / A$ is a $Σ$ -group, then $G$ is a $Σ$ -group if and only if $A$ is a $Σ$ -subgroup in $G$ provided that $A$ is equipped with a valuation induced by the restricted height function on $G$ . In particular, if in addition $A$ is pure in $G$ , $G$ is a $Σ$ -group precisely when $A$ is a $Σ$ -group. This extends the classical Dieudonné criterion (Portugal. Math., 1952) as well as it supplies our recent results in (Arch. Math. Brno, 2005), (Bull. Math. Soc....

Cellularity of a space of subgroups of a discrete group

Arkady G. Leiderman, Igor V. Protasov (2008)

Commentationes Mathematicae Universitatis Carolinae

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Given a discrete group $G$ , we consider the set $ℒ (G)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $ℒ (G)$ into the Cantor cube ${0, 1}^{G}$ . We show that the cellularity $c (ℒ (G)) \leq ℵ_{0}$ for every abelian group $G$ , and, for every infinite cardinal $τ$ , we construct a group $G$ with $c (ℒ (G)) = τ$ .

On topological groups with a small base and metrizability

Saak Gabriyelyan, Jerzy Kąkol, Arkady Leiderman (2015)

Fundamenta Mathematicae

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A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $U_{α} : α \in ℕ^{ℕ}$ , such that $U_{α} \subset U_{β}$ whenever β ≤ α for all $α, β \in ℕ^{ℕ}$ . The class of all metrizable topological groups is a proper subclass of the class $T G_{}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G \in T G_{}$ is metrizable, and hence G is strictly angelic. We deduce from...