极大等变紧化
Maximal equivariant compactifications
摘要：
设$G$是一个局部紧群。那么对于每个$G$ -空间$X$，最大$G$ -邻近$β_G$可以由最大拓扑邻近$β$表征如下：$$ A \ \overline{β_G} \ B \Leftrightarrow \exists V \in N_e \ \ \ VA \ \overlineβ \ VB. $$这里，$β_G \colon X \to β_G X$是$X$的最大$G$ -紧化（它是局部紧$G$的嵌入），$V$是$e$的邻域，而$A \ \overline{β_G} \ B$表示$A$和$B$的闭包在$β_G X$中不相交.注意$G$的局部紧性是必不可少的。这个定理是一个推论的一般结果最大的$\mathcal{U}$ -一致$G$ -紧化的一个有用的广泛的一类一致结构$\mathcal{U}$上的$G$ -空间不一定是局部紧群$G$。它特别有助于得出以下结果。设$(\mathbb{U}_1,d)$是Urysohn球面，$G=Iso(\mathbb{U}_1,d)$是它的具有点态拓扑的等距群.那么对于$\mathbb{U}_1$中的每对子集$A,B$，我们有$$ A \ \overline{β_G} \ B \Leftrightarrow \exists V \in N_e \ \ \ d(VA,VB) > 0. $$更一般地，对于任何$\aleph_0$ -分类度量$G$ -结构$(M,d)$也是如此，其中$G:=Aut(M)$是它的自同构群。
英文摘要：
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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