Maximal equivariant compactifications

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Abstract

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 ℵ0-categorical metric G-structure (M,d), where G:=Aut(M) is its automorphism group.

Original languageEnglish
Article number108372
JournalTopology and its Applications
Volume329
DOIs
StatePublished - 15 Apr 2023

Bibliographical note

Publisher Copyright:
© 2022 Elsevier B.V.

Keywords

  • Equivariant compactification
  • Gurarij sphere
  • Linearly ordered space
  • Proximity space
  • Thompson's group
  • Uniform space
  • Urysohn sphere

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