A Factorization Theorem for Topological Abelian Groups (original) (raw)
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Here “group” means additive abelian group. A compact group G contains δ–subgroups, that is, compact totally disconnected subgroups Δ such that G/Δ is a torus. The canonical subgroup Δ(G) of G that is the sum of all δ–subgroups of G turns out to have striking properties. Lewis, Loth and Mader obtained a comprehensive description of Δ(G) when considering only finite dimensional connected groups, but even for these, new and improved results are obtained here. For a compact group G, we prove the following: Δ(G) contains tor(G), is a dense, zero-dimensional subgroup of G containing every closed totally disconnected subgroup of G, and G/Δ(G) is torsion-free and divisible; Δ(G) is a functorial subgroup of G, it determines G up to topological isomorphism, and it leads to a “canonical” resolution theorem for G. The subgroup Δ(G) appeared before in the literature as td(G) motivated by completely different considerations. We survey and extend earlier results. It is shown that td, as a functor,...
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arXiv (Cornell University), 2008
For an abelian topological group G let G^* be the dual group of all continuous characters endowed with the compact open topology. A subgroup D of G determines G if the restriction homomorphism G^* --> D^* of the dual groups is a topological isomorphism. Given a scattered compact subset X of an infinite compact abelian group G such that |X|<w(G) and an open neighbourhood U of 0 in the circle group, we show that the set of all characters which send X into U has the same size as G^*. (Here w(G) denotes the weight of G.) As an application, we prove that a compact abelian group determined by its countable subgroup must be metrizable. This gives a negative answer to questions of Comfort, Hernandez, Macario, Raczkowski and Trigos-Arrieta, as well as provides short proofs of main results established in three manuscripts by these authors.
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Journal of Group Theory, 2000
We prove that every dense subgroup of a topological abelian group has the same 'convergence dual' as the whole group. By the 'convergence dual' we mean the character group endowed with the continuous convergence structure. We draw as a corollary that the continuous convergence structure on the character group of a precompact group is discrete and therefore a non-compact precompact group is never reflexive in the sense of convergence. We do not know if the same statement holds also for reflexivity in the sense of Pontryagin; at least in the category of metrizable abelian groups it does.
On characterized subgroups of compact abelian groups
Topology and its Applications, 2013
Let X be a compact abelian group. A subgroup H of X is called characterized if there exists a sequence u = (u n) of characters of X such that H = s u (X), where s u (X) := {x ∈ X: (u n , x) → 0 in T}. Every characterized subgroup is an F σδ-subgroup of X. We show that every G δ-subgroup of X is characterized. On the other hand, X has non-characterized F σ-subgroups. A subgroup H of X is said to be countable modulo compact (CMC) if H has a subgroup K such that it is a compact G δ-subgroup of X and H/K is countable. It is proved that every characterized subgroup H of X is CMC if and only if X has finite exponent. This result gives a complete description of the characterized subgroups of compact abelian groups of finite exponent. For every sequence u = (u n) of characters of X we define a refinement X u of X, that is a Čech complete locally quasi-convex (almost metrizable) group. With the sequence u we associate the closed subgroup H u of X u and the natural projection π X : X u → X such that π X (H u) = s u (X). This provides a description of the characterized subgroups of arbitrary compact abelian groups, extending the previously existing result from [25]. This description is new even in the case of metrizable compact groups.
Characterizing subgroups of compact abelian groups
Journal of Pure and Applied Algebra, 2007
We prove that every countable subgroup of a compact metrizable abelian group has a characterizing set. As an application, we answer several questions on maximally almost periodic (MAP) groups and give a characterization of the class of (necessarily MAP) abelian topological groups whose Bohr topology has countable pseudocharacter.