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Displaying similar documents to “On zero-dimensionality of subgroups of locally compact 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 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 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...

Subgroups of ℝ -factorizable groups

Constancio Hernández, Mihail G. Tkachenko (1998)

Commentationes Mathematicae Universitatis Carolinae

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The properties of ℝ -factorizable groups and their subgroups are studied. We show that a locally compact group G is ℝ -factorizable if and only if G is σ -compact. It is proved that a subgroup H of an ℝ -factorizable group G is ℝ -factorizable if and only if H is z -embedded in G . Therefore, a subgroup of an ℝ -factorizable group need not be ℝ -factorizable, and we present a method for constructing non- ℝ -factorizable dense subgroups of a special class of ℝ -factorizable groups. Finally, we construct...

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 ) ≤ 2 τ · in ( H ) and c ( ( H ) τ ) ≤ 2 2 τ · in ( H ) hold for every topological group H and every cardinal τ ≥ 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...

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 ) ) ≤ ℵ 0 for every abelian group G , and, for every infinite cardinal τ , we construct a group G with c ( ℒ ( G ) ) = τ .