论文标题
适当的度量空间的粗略紧凑
Coarse compactifications of proper metric spaces
论文作者
论文摘要
本文研究了粗糙的紧凑型及其边界。我们将两个替代描述引入了ROE对粗磨的原始定义。一种方法在$ x $上使用有界函数，可以扩展到边界。当紧凑型粗糙时，它们准确地满足了希格森的特性。另一种方法定义了$ x $的子集的关系，该子集告诉边界上两个子集何时封闭。当这种关系定义粗糙的紧凑型时，一组公理会表征。这种关系称为大规模接近。基于这项基础工作，我们研究了希格森紧凑型，弗洛伊达尔紧凑和gromov紧凑型的粗糙紧凑型示例。对于每个示例，我们表征有界函数，可以扩展到粗糙的紧凑型和相应的大规模接近关系。我们为属性提供了另一种证据，即希格森紧凑型在粗糙的紧凑型之间是普遍的。此外，在具有完全断开边界的粗糙紧凑型之间，友式压实是普遍的。如果$ x $是双曲线测量适中，则有一个封闭的嵌入$ν（\ Mathbb r _+）\ times \ times \ partial x \ toν（x）$。如果$ x $是一棵树，它的图像是$ν（x）$的缩回。
This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on $X$ that can be extended to the boundary. They satisfy the Higson property exactly when the compactification is coarse. The other approach defines a relation on subsets of $X$ which tells when two subsets closure meet on the boundary. A set of axioms characterizes when this relation defines a coarse compactification. Such a relation is called large-scale proximity. Based on this foundational work we study examples for coarse compactifications Higson compactification, Freudenthal compactification and Gromov compactification. For each example we characterize the bounded functions which can be extended to the coarse compactification and the corresponding large-scale proximity relation. We provide an alternative proof for the property that the Higson compactification is universal among coarse compactifications. Furthermore the Freudenthal compactification is universal among coarse compactifications with totally disconnected boundary. If $X$ is hyperbolic geodesic proper then there is a closed embedding $ν(\mathbb R_+)\times \partial X\to ν(X)$. Its image is a retract of $ν(X)$ if $X$ is a tree.
