可度量化的一致空间
Metrizable uniform spaces
摘要：
一般拓扑学的三个主题：商空间、绝对收缩核和逆极限--在可度量化的一致空间中重新讨论，着眼于几何和代数拓扑学中的应用。结果包括：1）如果f：A -> Y是一致连续映射，其中X和Y是度量空间，A是X的闭子集，我们证明了具有商一致性的邻接空间X \cup_f Y（因此也具有其拓扑）是可通过显式度量度量化的.这就产生了在可度量化的一致空间范畴中锥、连接和映射柱的自然构造，我们证明了这些构造与基于子空间（赋范线性空间）、乘积（与锥）和l_2度量的各向同性的构造是一致的。2)我们重新审视Isbell的一致ANRs理论，Garg和Nhu在可度量化的情况下进行了改进。证明了点紧多面体P的迭代闭路空间Ω^n P是一致ANR。给出了可度量化一致空间X中一致ANR的4个刻画：（i）X的完备化是一致ANR，余项一致是完备化中的Z-集;（ii）X是一致局部可收缩的，满足Hahn逼近性质;（iii）X是一致ε-同伦，对每个ε>0，X都被一致ANR控制;（iv）X是具有“近分裂”键映射的一致ANR的逆极限。几个章节主要用于阐述：（一）介绍一致空间，重点是可度量的情况下;（五）空间的可测函数;（六）空间的概率测度和其他测度空间。
英文摘要：
Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spaces and A is a closed subset of X, we show that the adjunction space X\cup_f Y with the quotient uniformity (hence also with the topology thereof) is metrizable, by an explicit metric. This yields natural constructions of cone, join and mapping cylinder in the category of metrizable uniform spaces, which we show to coincide with those based on subspace (of a normed linear space); on product (with a cone); and on the isotropy of the l_2 metric. 2) We revisit Isbell's theory of uniform ANRs, as refined by Garg and Nhu in the metrizable case. The iterated loop spaces Ω^n P of a pointed compact polyhedron P are shown to be uniform ANRs. Four characterizations of uniform ANRs among metrizable uniform spaces X are given: (i) the completion of X is a uniform ANR, and the remainder is uniformly a Z-set in the completion; (ii) X is uniformly locally contractible and satisfies the Hahn approximation property; (iii) X is uniformly ε-homotopy dominated by a uniform ANR for each ε>0; (iv) X is an inverse limit of uniform ANRs with "nearly splitting" bonding maps. Several chapters are devoted primarily to exposition: (I) an introduction to uniform spaces, with a focus on the metrizable case; (V) the space of measurable functions; (VI) the space of probability measures and other measure spaces.
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