所有尺度下的零维
Dimension zero at all scales
摘要：
我们考虑的概念的尺寸在四个类别：（无界）可分的度量空间和（度量适当的）Lipschitz映射的类别，和（无界）可分的度量空间和（度量适当的）一致映射的类别。对大尺度维数和小尺度维数作了统一的处理。我们证明了在所有范畴中，一个空间的维数为零当且仅当它等价于一个超度量空间。此外，0维空间的特征在于通过收缩到子空间。在所有的范畴中都存在一个普遍的零维空间。在Lipschitz范畴中，零维空间通过单位0-球面的映射扩张来刻画。任何渐近维数为零的可数群粗略地等价于循环群的直和。我们构造了无数个粗糙不等价超度量空间的例子。
英文摘要：
We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale dimension. We show that in all categories a space has dimension zero if and only if it is equivalent to an ultrametric space. Also, 0-dimensional spaces are characterized by means of retractions to subspaces. There is a universal zero-dimensional space in all categories. In the Lipschitz Category spaces of dimension zero are characterized by means of extensions of maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is coarsely equivalent to a direct sum of cyclic groups. We construct uncountably many examples of coarsely inequivalent ultrametric spaces.
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