Maximal equivariant compactifications

Let $G$ be a locally compact group. Then for every $G$-space $X$ the maximal $G$-proximity ${\beta }_G$ can be characterized by the maximal topological proximity $\beta$ as follows: $$A\overline{{\beta }_G}B\iff \mathrm{\exists }V\in N_{e}VA\bar{\beta }VB\mathrm{.}$$ Here, $\beta_G \colon X \to \beta_G X$isthemaximal$G$-compactificationof$X$ (which is an embedding for locally compact $G$), $V$ is a neighborhood of $e$ and $A
\overline{\beta_G} \ B$meansthattheclosuresof$A$and$B$donotmeetin$\beta_G X$\mathrm{.}Notethatthelocalcompactnessof$G$ 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 $$A\overline{{\beta }_G}B\iff \mathrm{\exists }V\in N_{e}d(VA,VB)>0.$$ More generally, the same is true for any ${\mathrm{\aleph }}_0$-categorical metric $G$-structure $(M,d)$, where $G:=Aut(M)$ is its automorphism group.