## Abstract

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_{e}VAβ‾VB. Here, β_{G}:X→β_{G}X 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_{G}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 (U_{1},d) be the Urysohn sphere and G=Iso(U_{1},d) is its isometry group with the pointwise topology. Then for every pair of subsets A,B in U_{1}, we have Aβ_{G}‾B⇔∃V∈N_{e}d(VA,VB)>0. More generally, the same is true for any ℵ_{0}-categorical metric G-structure (M,d), where G:=Aut(M) is its automorphism group.

Original language | English |
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Article number | 108372 |

Journal | Topology and its Applications |

Volume | 329 |

DOIs | |

State | Published - 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