? 2022 Elsevier B.V.Let G be a locally compact group. Then for every G-space X the maximal G-proximity βG can be characterized by the maximal topological proximity β as follows: AβG￣B??V∈NeVAβ￣VB. Here, βG:X→βGX 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 AβG￣B means that the closures of A and B do not meet in βGX. 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 AβG￣B??V∈Ned(VA,VB)>0. More generally, the same is true for any ?0-categorical metric G-structure (M,d), where G:=Aut(M) is its automorphism group.
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