Topological and uniform structures on universal covering spaces

Abstract

We discuss various uniform structures and topologies on the universal covering space $\tilde{X}$ and on the fundamental group ${\pi }_1(X,x_0)$. We introduce a canonical uniform structure $CU(X)$ on a topological space $X$ and use it to relate topologies on $\tilde{X}$ 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.

preliminary version

References in corpus (4)

• The fundamental group as a topological group
• Partitions of unity
• Homotopical Smallness and Closeness
• The Hawaiian earring group and metrizability

Cited by in corpus (8)

• 1-Dimensional Intrinsic Persistence of Geodesic Spaces
• Anomalous coverings
• On Topologized Fundamental Group and covering spaces of topological groups
• Fibrations, unique path lifting, and continuous monodromy
• On Generalized Covering Groups of Topological Groups
• On Strong Small Loop Transfer Spaces Relative to Subgroups of Fundamental Groups
• Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups
• On Subgroups of Topologized Fundamental Groups and Generalized Coverings