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Let F = ind lim Fn be an infinite-dimensional LF-space with density dens F = τ (≥ ℵ0) such that some Fn is infinite-dimensional and dens Fn = τ. It is proved that every open subset of F is homeomorphic to the product of an 2(τ)-manifold and R ∞ = ind lim R n (hence the product of an open subset of 2(τ) and R ∞). As a consequence, any two open sets in F are homeomorphic if they have the same homotopy type.

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We prove that a topological group G is (locally) homeomorphic to an LF-space if G = n∈ω Gn for some increasing sequence of subgroups (Gn)n∈ω such that (1) for any neighborhoods Un ⊂ Gn, n ∈ ω, of the neutral element e ∈ Gn ⊂ G, the set(2) each group Gn is (locally) homeomorphic to a Hilbert space;(3) for every n ∈ N the quotient map Gn → Gn/G n−1 is a locally trivial bundle; (4) for infinitely many numbers n ∈ N each Z-point in the quotient space Gn/G n−1 = {xG n−1 : x ∈ Gn} is a strong Z-point.2010 Mathematics Subject Classification. 57N20; 46A13; 22A05.

Abstract. Let TB be the category of totally bounded, locally compact metric spaces with the C coarse structures. We show that if X and Y are in TB then X and Y are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories TB → K, where K is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space X induced by some metrizable compacti cationX is determined only by the topology of the remainderX ∖ X.