粗糙结构的另一种定义
An alternative definition of coarse structures
摘要：
John Roe \cite{Roe lectures}通过考虑$X\times X$的子集引入了任意集合$X$的粗糙结构。这个定义对分析家来说是自然的，但对拓扑学家和几何学家来说就有点难以理解了。本文通过一致有界族的概念引入了$X$上的大尺度结构，并证明了它们与$X$上的粗结构的等价性.这样，大尺度几何的所有基本概念（渐近维数、慢振荡函数、希格森紧化）都有了自然的定义，度量几何的基本结果也就转移到了粗几何中。
英文摘要：
John Roe \cite{Roe lectures} introduced coarse structures for arbitrary sets $X$ by considering subsets of $X\times X$. That definition, while natural for analysts, is a bit more difficult to digest for topologists and geometers. In this paper we introduce large scale structures on $X$ via the notion of uniformly bounded families and we show their equivalence to coarse structures on $X$. That way all basic concepts of large scale geometry (asymptotic dimension, slowly oscillating functions, Higson compactification) have natural definitions and basic results from metric geometry carry over to coarse geometry.
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