  In topology, the notions of the fundamental group and the universal cover are
closely intertwined. By importing usual notions from topology into the
algebraic and arithmetic setting, we construct a fundamental group family from
a universal cover, both of which are schemes. A geometric fiber of the
fundamental group family (as a topological group) is canonically the 'etale
fundamental group. The constructions apply to all connected quasicompact
quasiseparated schemes. With different methods and hypotheses, this fundamental
group family was already constructed by Deligne.
