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We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if $H$ is a connected locally compact Abelian subgroup of a Hausdorff topological group $G$ and the quotient space $G / H$ is sequentially connected, then so is $G$ .

Construction techniques for some thin sets in duals of compact abelian groups

D. J. Hajela (1986)

Annales de l'institut Fourier

Various techniques are presented for constructing $Λ$ (p) sets which are not $Λ (p + ϵ)$ for all $ϵ > 0$ . The main result is that there is a $Λ$ (4) set in the dual of any compact abelian group which is not $Λ (4 + ϵ)$ for all $ϵ > 0$ . Along the way to proving this, new constructions are given in dual groups in which constructions were already known of $Λ$ (p) not $Λ (p + ϵ)$ sets, for certain values of $p$ . The main new constructions in specific dual groups are:– there is a $Λ$ (2k) set which is not $Λ (2 k + ϵ)$ in $Z (2) \oplus Z (2) \oplus \dots$ for all $2 \leq k$ , $k \in N$ and $ϵ > 0$ , and in $Z (p) \oplus Z (p) \oplus \dots$ ( $p$ a prime,...

Der Satz von Helson-Reiter für spezielle nilopotente Lie-Gruppen.

H.J. Hey, J. Ludwig (1979)

Mathematische Annalen

Dual algebras.

Martin E. Walter (1986)

Mathematica Scandinavica

Dualité, et transformation de Fourier $p$ -adiques

Bernard de Mathan (1971/1972)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Dualité et transformation de Fourier p-adiques

Bernard de MATHAN de (1971/1972)

Seminaire de Théorie des Nombres de Bordeaux

Duality, imprimitivity, reciprocity

Krzysztof Maurin, Lidia Maurin (1975)

Annales Polonici Mathematici

Duals of Orbit Spaces in Groups With Relatively Compact Inner Automorphism Groups Are Hypergroups.

R.W. Henrichs, K. Hartmann, R. Lasser (1979)

Monatshefte für Mathematik

Extension de la catégorie des algèbres de Kac

M. Enock, J. M. Schwartz (1986)

Annales de l'institut Fourier

On munit la classe des algèbres de Kac d’une nouvelle classe de morphismes, stable par dualité. Cela permet de rendre compte, dans les cas abélien ou symétrique, de la catégorie des groupes localement compacts munis des morphismes continus de groupe. Le lien avec les morphismes précédemment définis et beaucoup plus restrictifs est établi.

Extensions of Pontryagin Duality.

Rangachari Venkataraman (1975)

Mathematische Zeitschrift

Induced representations and duality results for commutative hypergroups.

Michael Voit, Peter Hermann (1995)

Forum mathematicum

Isolated Points in Duals of Certain Locally Compact Groups.

Terje Sund (1976)

Mathematische Annalen

Mapping properties of characters of LCA groups

D. Armacost (1972)

Fundamenta Mathematicae

Medias en espacios localmente convexos y semi-reflexivilidad.

F. Bombal, G. Vera (1973)

Collectanea Mathematica

Metrization criteria for compact groups in terms of their dense subgroups

Dikran Dikranjan, Dmitri Shakhmatov (2013)

Fundamenta Mathematicae

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 by all its...

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