  Considering non-constant holomorphic maps $\beta_{i}:S_{i}\to S_{0}$,
$i\in\{1,2\}$, between non-compact Riemann surfaces for which it is associated
its fiber product $S_{1}\times_{(\beta_{1},\beta_{2})}S_{2}$. With this
setting, in this paper we relate the ends space of such fiber product to the
ends space of its normal fiber product. Moreover, we provide conditions on the
maps $\beta_{1}$ and $\beta_{2}$ to guarantee connectednes on the fiber
product. From these conditions, we link the ends space of fiber product with
the topological type of the Riemann surfaces $S_{1}$ and $S_{2}$. We also study
the fiber product over infinite hyperelliptic curves and discuss its
connectedness and ends space.
