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Displaying similar documents to “Valdivia compact abelian groups.”

Isomorphisms of Direct Products of Finite Cyclic Groups

Kenichi Arai, Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

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In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

The Bohr compactification, modulo a metrizable subgroup

W. Comfort, F. Trigos-Arrieta, S. Wu (1993)

Fundamenta Mathematicae

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The authors prove the following result, which generalizes a well-known theorem of I. Glicksberg [G]: If G is a locally compact Abelian group with Bohr compactification bG, and if N is a closed metrizable subgroup of bG, then every A ⊆ G satisfies: A·(N ∩ G) is compact in G if and only if {aN:a ∈ A} is compact in bG/N. Examples are given to show: (a) the asserted equivalence can fail in the absence of the metrizability hypothesis, even when N ∩ G = {1}; (b) the asserted equivalence can...

Imposing psendocompact group topologies on Abeliau groups

W. Comfort, I. Remus (1993)

Fundamenta Mathematicae

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The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, m ( α ) ≤ 2 α . We show:    Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m ( α ) ≤ r 0 ( G ) ≤ γ ≤ 2 α , or α > ω and α ω ≤ r 0 ( G ) ≤ 2 α , then G admits a pseudocompact group topology of weight α.  Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies r 0 ( G ) ≥ m ( α ) .  Theorem 5.2(b). If G is divisible Abelian with 2 r 0 ( G ) ≤ γ , then G admits...