  We discuss various uniform structures and topologies on the universal
covering space $\widetilde X$ 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 $\widetilde X$ 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.
