We introduce the moduli space of marked, complete, Nielsen-convex hyperbolic structures on a surface of negative, but not necessarily finite, Euler characteristic. The emphasis is on the case in which the surface is of infinite type, the aim being to study the mapping class group of such a surface via its action on this marked moduli space. We define a topology on the marked moduli space, in multiple ways. This marked moduli space reduces to the usual Teichmüller space for finite type surfaces. Since a big mapping class group is a topological group with a nontrivial topology, a basic question is whether its action on the marked moduli space is continuous. We answer this question in the affirmative. We also show that the marked moduli space is contractible.