On the Chabauty space of locally compact abelian groups

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

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