  We investigate the thermalization of a two-component scalar field across a
second-order phase transition under extremely fast quenches. We find that
vortices start developing at the final temperature of the quench, i.e., below
the critical point. Specifically, we find that vortices emerge once the
fluctuating field departures from its symmetric state and evolves towards a
metastable and inhomogenous configuration. The density of primordial vortices
at the relaxation time is a decreasing function of the final temperature of the
quench. Subsequently, vortices and antivortices annihilate at a rate that
eventually determines the total thermalization time. This rate decreases if the
theory contains a discrete anisotropy, which otherwise leaves the primordial
vortex density unaffected. Our results thus establish a link between the
topological processes involved in the vortex dynamics and the delay in the
thermalization of the system.
