  James \cite{Jam} introduced uniform covering maps as an analog of covering
maps in the topological category. Subsequently Berestovskii and Plaut
\cite{BP3} introduced a theory of covers for uniform spaces generalizing their
results for topological groups \cite{BP1}-\cite{BP2}. Their main concepts are
discrete actions and pro-discrete actions, respectively. In case of
pro-discrete actions Berestovskii and Plaut provided an analog of the universal
covering space and their theory works well for the so-called coverable spaces.
As will be seen in Section \ref{SECTION-Comparison}, \cite{BP3} generalizes
only regular covering maps in topology and pro-discrete actions may not be
preserved by compositions.
  In this paper we redefine the uniform covering maps and we generalize
pro-discrete actions using Rips complexes and the chain lifting property. We
expand the concept of generalized paths of Krasinkiewicz and Minc
\cite{KraMin}.
