1,233 research outputs found

    On the Chabauty space of locally compact abelian groups

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    This paper contains several results about the Chabauty space of a general locally compact abelian group. Notably, we determine its topological dimension, we characterize when it is totally disconnected or connected; we characterize isolated points.Comment: 24 pages, 0 figur

    On the subset Combinatorics of G-spaces

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    Let GG be a group and let XX be a transitive GG-space. We classify the subsets of XX with respect to a translation invariant ideal J\mathcal{J} in the Boolean algebra of all subsets of XX, introduce and apply the relative combinatorical derivations of subsets of XX. Using the standard action of GG on the Stone-Cˇ\check{C}ech compactification βX\beta X of the discrete space XX, we characterize the points pβXp\in\beta X isolated in GpGp and describe a size of a subset of XX in terms of its ultracompanions in βX\beta X. We introduce and characterize scattered and sparse subsets of XX from different points of view

    Ultrafilters on GG-spaces

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    For a discrete group GG and a discrete GG-space XX, we identify the Stone-\v{C}ech compactifications βG\beta G and βX\beta X with the sets of all ultrafilters on GG and XX, and apply the natural action of βG\beta G on βX\beta X to characterize large, thick, thin, sparse and scattered subsets of XX. We use GG-invariant partitions and colorings to define GG-selective and GG-Ramsey ultrafilters on XX. We show that, in contrast to the set-theoretical case, these two classes of ultrafilters are distinct. We consider also universally thin ultrafilters on ω\omega, the TT-points, and study interrelations between these ultrafilters and some classical ultrafilters on ω\omega