万有覆叠空间上的拓扑结构与一致结构
Topological and uniform structures on universal covering spaces
摘要：
讨论了万有覆叠空间$\widetilde X$和基本群$π_1(X,x_0)$上的各种一致结构和拓扑.本文在拓扑空间$X$上引入了一个规范一致结构$CU(X)$，并用它来联系$\widetilde X$上的拓扑和$\widetilde{CU(X)}$上的一致结构.使用我们的概念，普遍皮亚诺空间，我们显示之间的连接的拓扑结构介绍了Spanier和统一的结构Berestovskii和Plaut。给出了Berestovskii-Plaut结构与$X$中路空间上一致收敛性结构所生成的Berestovski-Plaut结构相同的一个充必要条件.我们还描述了当拓扑的Spanier是相同的商的紧开拓扑的道路空间。
英文摘要：
We discuss various uniform structures and topologies on the universal covering space $\widetilde X$ and on the fundamental group $π_1(X,x_0)$. We introduce a canonical uniform structure $CU(X)$ on a topological space $X$ and use it to relate topologies on $\widetilde X$ and uniform structures on $\widetilde{CU(X)}$. Using our concept of universal Peano space we show connections between the topology introduced by Spanier and a uniform structure of Berestovskii and Plaut. We give a sufficient and necessary condition for Berestovskii-Plaut structure to be identical with the one generated by the uniform convergence structure on the space of paths in $X$. We also describe when the topology of Spanier is identical with the quotient of the compact-open topology on the space of paths.
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