叠加和群作用
Overlays and group actions
摘要：
Overlay是由R. H. Fox [6]作为覆盖映射的子类。我们提供了一个不同的观点覆盖：它类似于仿紧空间的定义，通过开覆盖的星加细。一个是为覆盖映射引入覆盖结构，如果$p:X\to Y$具有星型细化的覆盖结构，则它是覆盖。我们证明了两个特征的覆盖：一个使用的存在性和唯一性的升降机的离散链，第二个地图诱导单纯覆盖的神经的某些覆盖。我们利用这些刻画改进了Eda-Matijevi\' c关于值域为紧拓扑群的覆盖域上的拓扑群结构的结果.在连通可度量化空间之间的满射映射$p:X\to Y$的情况下，我们将覆盖刻画为局部等距：$p:X\to Y$是覆盖当且仅当可以度量化$X$和$Y$，使得$p|B(x,1):B(x,1)\to B(p(x),1)$是每个$x\in X$的等距。
英文摘要：
Overlays were introduced by R. H. Fox [6] as a subclass of covering maps. We offer a different view of overlays: it resembles the definition of paracompact spaces via star refinements of open covers. One introduces covering structures for covering maps and $p:X\to Y$ is an overlay if it has a covering structure that has a star refinement. We prove two characterizations of overlays: one using existence and uniqueness of lifts of discrete chains, the second as maps inducing simplicial coverings of nerves of certain covers. We use those characterizations to improve results of Eda-Matijevi\' c concerning topological group structures on domains of overlays whose range is a compact topological group. In case of surjective maps $p:X\to Y$ between connected metrizable spaces we characterize overlays as local isometries: $p:X\to Y$ is an overlay if and only if one can metrize $X$ and $Y$ in such a way that $p|B(x,1):B(x,1)\to B(p(x),1)$ is an isometry for each $x\in X$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
