  We consider Riemann surfaces obtained from nodal curves with infinite
cylinders in the place of nodal and marked points, and study the space of
finite energy vortices defined on these surfaces. To compactify the space of
vortices, we need to consider stable vortices - these incorporate breaking of
cylinders and sphere bubbling in the fibers. In this paper, we prove that the
space of gauge equivalence classes of stable vortices representing a fixed
equivariant homology class is compact and Hausdorff under the Gromov topology.
We also show that this space is homeomorphic to the moduli space of quasimaps
defined by Ciocan-Fontanine, Kim and Maulik.
