Topologies on groups determined by discrete subsets

Topologies on IR” induced by smooth subsets

Ancel, F.D., Topologies on Iw” induced by smooth subsets, Topology and its Applications 43 (1992) 189-201. If Y is a collection of subsets of iw”, let Y!? denote the largest topology on Iw” which restricts to the standard topology on each element of Y, and let ?t$ denote the homeomorphism group of Iw” wtth the topology &. Let Y_, denote the standard topology on Iw” and let %?_d denote the homeo...

Topologies on Spaces of Subsets

Topologies Generated by Discrete Subspaces

A topological space X is called discretely generated if for every subset A⊂X we have A=∪{D:D⊂A and D is a discrete subspace of X}. We say that X is weakly discretely generated if A⊂X and A6=A implies D\A6=∅ for some discrete D⊂A. It is established that sequential spaces, monotonically normal spaces and compact countably tight spaces are discretely generated. We also prove that every compact spa...

On Group Topologies Determined by Families of Sets

Let G be an abelian group, and F a downward directed family of subsets of G. In [6], I. Protasov and E. Zelenyuk describe the finest group topology T on G under which F converges to 0; in particular, their description yields a criterion for T to be Hausdorff. They then show that if F is the filter of cofinite subsets of a countable subset X ⊆ G (the Fréchet filter on X), there is a simpler crit...

On radii of spheres determined by subsets of Euclidean space

In this paper we consider the problem that how large the Hausdorff dimension of E ⊂ Rd needs to be to ensure the radii set of (d − 1)-dimensional spheres determined by E has positive Lebesgue measure. We obtain two results. First, by extending a general mechanism for studying Falconer-type problems in [4], we prove that it holds when dimH(E) > d− 1+ 1 d and in R2, the index 3 2 is sharp for thi...