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The dual group of a dense subgroup

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

Czechoslovak Mathematical Journal

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 group is...

The existence of initially $ω_{1}$ -compact group topologies on free Abelian groups is independent of ZFC

Artur Hideyuki Tomita (1998)

Commentationes Mathematicae Universitatis Carolinae

It was known that free Abelian groups do not admit a Hausdorff compact group topology. Tkachenko showed in 1990 that, under CH, a free Abelian group of size $ℭ$ admits a Hausdorff countably compact group topology. We show that no Hausdorff group topology on a free Abelian group makes its $ω$ -th power countably compact. In particular, a free Abelian group does not admit a Hausdorff $p$ -compact nor a sequentially compact group topology. Under CH, we show that a free Abelian group does not admit a Hausdorff...

Displaying 1 – 9 of 9

The dual group of a dense subgroup

The existence of initially $ω_{1}$ -compact group topologies on free Abelian groups is independent of ZFC

The Lie algebra of a nuclear group.

The size of sums of sets

The structure of L-ideals of measures algebras

The Witt group of quadratic characters.

Three results on I-sets

Topologisch Linksengelsche Elemente.

Towards a Galois theory for crossed products of C*-algebras.

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