2022/01/31 by Megrelishvili, Michael · 1 citation
#54D35 #54F05 #54H15 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)
paper · doi:10.48550/arxiv.2201.13426
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: A βG B ⇔ ∃ V ∈ Ne VA β VB. Here, βG \colon X → βG X is the maximal G-compactification of X (which is an embedding for locally compact G), V is a neighborhood of e and A βG B means that the closures of A and B do not meet in βG X. 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 (\mathbbU1,d) be the Urysohn sphere and G=Iso(\mathbbU1,d) is its isometry group with the pointwise topology. Then for every pair of subsets A,B in \mathbbU1, we have A βG B ⇔ ∃ V ∈ Ne d(VA,VB) gt; 0. More generally, the same is true for any ℵ0-categorical metric G-structure (M,d), where G:=Aut(M) is its automorphism group.