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This is true if $F_n$ is a right Følner sequence, i.e., $\frac{|F_n \Delta F_ng |}{|F_n|} \to 0$ for all $g \in G$. Indeed, if $F \subset G$ is finite, then the condition $g F \cap S \not= \emptyset$, for every $g \in G$, is equivalent to the condition $S F^{-1} = G$. Hence $D^*(S) = \frac{1}{|F|} \sum_{f \in F} D^* (Sf^{-1}) \geq \frac{1}{|F|} D^*(S ... 2 Abelian groups are exactly$Z$-modules ($Z$- the integers), so it would be better to write$A\otimes_ZR$for this. Then$A\mapsto A\otimes_Z R$which the topology you look for is the left adjoint functor from the category of abelian topological groups to the category of topological vector spaces to the forgetful functor which associates to each topological ... 3 I think I can show that$H \neq K$. We will think about functions that can be constructed using the group operations and constant elements of the group, like$f(y_1,y_2,y_3) = x_4^2 y_1 y_2 x_3^{-1} y_3$. Consider the following topology: A set$U$is open if, for each$k$-variable function$f$such that$f(1,\dots, 1) \in U$, there is a constant$N$such ... 4 A similar question was answered here: http://mathoverflow.net/a/119962/2926. The idea is to start with your favorite non-trivial abelian group$A$, say$A = \mathbb{Z}/(2)$(viewed as a group object in$\mathbf{Set}$) and apply to it a sequence of product-preserving functors$\$\mathbf{Set} \stackrel{K}{\to} \mathbf{Cat} \stackrel{\text{nerve}}{\to} ...