Maximal equivariant compactifications

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