论文标题
最大的模棱两可的压缩
Maximal equivariant compactifications
论文作者
论文摘要
令$ g $为本地紧凑型组。然后，对于每一个$ g $ -space $ x $，最大$ g $ -proximity $β_g$都可以由最大拓扑近端$β$来表征：$$ a \ a \\ overline {β_g} {β_g} \ b \ leftrightArlow \ v \ v \ v \ v \ v \ v va $$ 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 $β_G X$.请注意，$ g $的本地紧凑性至关重要。该定理是作为最大$ \ MATHCAL {U} $ - 均匀$ G $ -Compactification的一般结果的必然结果，用于在$ G $ -SPACES上使用$ G $ spaces上的有用的宽类均匀结构$ \ MATHCAL {U} $。它特别有助于得出以下结果。令$（\ mathbb {u} _1，d）$为Urysohn Sphere，$ G = ISO（\ Mathbb {U} _1，d）$是其具有点式拓扑的等轴测组。然后，对于每对子集$ a，b $ in $ \ mathbb {u} _1 $，我们都有$$ a \ \ edline {β_g} \ b \ b \ leftrightArrow \ esists v \ en_e \ \ \ \ \ \ \ \ \ d（va，va，va，va，va，va，va）> 0. $ \ a $ cy n $ cy eleptor， $ g $ - 结构$（m，d）$，其中$ g：= aut（m）$是其自动形态组。
Let $G$ be a locally compact group. Then for every $G$-space $X$ the maximal $G$-proximity $β_G$ can be characterized by the maximal topological proximity $β$ as follows: $$ 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.
