统一大尺度和小尺度几何
Unifying large scale and small scale geometry
摘要：
集合$X$上的拓扑与投影（即幂等线性算子）$cl:2^X\to 2^X$相同，它对所有$A\subset X$都满足$A\subset cl(A)$。这是一个很好的方式来总结库拉托夫斯基的闭包运算符。集合$X$上的基本几何是点积$\cdot:2^X\times 2^X\to 2^Y$。它的等价形式是$X$子集上的正交关系。最佳情况是如果正交关系满足从线性代数的平行-垂直分解的变体。我们表明，这一概念统一的小规模（拓扑，邻近空间，一致空间）和大规模（粗空间，大规模空间）。利用正交关系，我们定义了大规模紧化，推广所有著名的紧化：希格森冠，格罗莫夫边界，Čech-Stone紧化，Samuel-Smirnov紧化，和Freudenthal紧化。
英文摘要：
A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set $X$ is a dot product $\cdot:2^X\times 2^X\to 2^Y$. Its equivalent form is an orthogonality relation on subsets of $X$. The optimal case is if the orthogonality relation satisfies a variant of parallel-perpendicular decomposition from linear algebra. We show that this concept unifies small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). Using orthogonality relations we define large scale compactifications that generalize all well-known compactifications: Higson corona, Gromov boundary, Čech-Stone compactification, Samuel-Smirnov compactification, and Freudenthal compactification.
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