We introduce a categorical closure operator g in the category of topological abelian groups (and continuous homomorphisms) as a Galois closure with respect to an appropriate Galois correspondence defined by means of the Pontryagin dual of the underlying group. We prove that a topological abelian group G is maximally almost periodic if and only if every cyclic subgroup of G is g-closed. This generalizes a property characterizing the circle group from (Studia Sci. Math. Hungar. 38 (2001) 97-113. A characterization of the circle group and the p-adic integers via sequential limit laws, preprint). and answers an appropriate version of a question posed in (A characterization of the circle group and the p-adic integers via sequential limit laws, preprint)
We study various weaker versions of metrizability for pseudocompact abelian groups G: singularity (G possesses a compact metrizable subgroup of the form mG, m>0), almost connectedness (G is metrizable modulo the connected component) and various versions of extremality in the sense of Comfort and co-authors (s-extremal, if G has no proper dense pseudocompact subgroups, r-extremal, if G admits no proper pseudocompact refinement). We introduce also weaklyextremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). It turns out that this “symmetric” version of ex-tremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining simpler uniform proofs of most of the known results. We characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. We give also a criterion for r-extremality for connected pseudocompact groups
For a recursively defined sequence u := (un) of integers, we describe the subgroup tu(T) of the elements x of the circle group T satisfying limn unx = 0. More attention is dedicated to the sequences satisfying a secondorder recurrence relation. In this case, we show that the size and the free-rank of tu(T) is determined by the asymptotic behaviour of the ratios qn = un u n−1 and we extend previous results of G. Larcher, C. Kraaikamp, and P. Liardet obtained from continued fraction expansion.
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