  Consider a connected topological space $X$ with a point $x \in X$ and let $K$
be a field with the discrete topology. We study the Tannakian category of
finite dimensional (flat) vector bundles on $X$ and its Tannakian dual $\pi_K
(X,x)$ with respect to the fibre functor in $x$. The maximal pro-\'etale
quotient of $\pi_K (X,x)$ is the \'etale fundamental group of $X$ studied by
Kucharczyk and Scholze. For well behaved topological spaces, $\pi_K (X,x)$ is
the pro-algebraic completion of the ordinary fundamental group $\pi_1 (X,x)$.
  We obtain some structural results on $\pi_K (X,x)$ by studying
(pseudo-)torsors attached to its quotients. This approach uses ideas of Nori in
algebraic geometry and a result of Deligne on Tannakian categories. We also
calculate $\pi_K (X,x)$ for some generalized solenoids.
