Maximal equivariant compactifications

Let G be a locally compact group. Then for every G-space X the maximal G-proximity β_G can be characterized by the maximal topological proximity β as follows: Aβ_G‾B⇔∃V∈N_eVAβ‾VB. Here, β_G:X→β_GX is the maximal G-compactification of X (which is an embedding for locally compact G by a classical result of J. de Vries), V is a neighbourhood of e and Aβ_G‾B means that the closures of A and B do not meet in β_GX. Note that the local compactness of G is essential. This theorem comes as a corollary of a general result about maximal U-uniform G-compactifications for a useful wide class of uniform structures U on G-spaces for not necessarily locally compact groups G. It helps, in particular, to derive the following result. Let (U₁,d) be the Urysohn sphere and G=Iso(U₁,d) is its isometry group with the pointwise topology. Then for every pair of subsets A,B in U₁, we have Aβ_G‾B⇔∃V∈N_ed(VA,VB)>0. More generally, the same is true for any ℵ₀-categorical metric G-structure (M,d), where G:=Aut(M) is its automorphism group.