Maximal equivariant compactifications

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: $A \ \overline{β_G} \ B \Leftrightarrow \exists V \in N_e \ \ \ VA \ \overlineβ \ VB.$ Here, $β_G \colon X \to β_G X$ is the maximal $G$-compactification of $X$ (which is an embedding for locally compact $G$), $V$ is a neighborhood of $e$ and $A \ \overline{β_G} \ B$ 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 $\mathcal{U}$-uniform $G$-compactifications for a useful wide class of uniform structures $\mathcal{U}$ on $G$-spaces for not necessarily locally compact groups $G$. It helps, in particular, to derive the following result. Let $(\mathbb{U}_1,d)$ be the Urysohn sphere and $G=Iso(\mathbb{U}_1,d)$ is its isometry group with the pointwise topology. Then for every pair of subsets $A,B$ in $\mathbb{U}_1$, we have $A \ \overline{β_G} \ B \Leftrightarrow \exists V \in N_e \ \ \ d(VA,VB) > 0.$ More generally, the same is true for any $\aleph_0$-categorical metric $G$-structure $(M,d)$, where $G:=Aut(M)$ is its automorphism group.

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