  Phase transitions are ubiquitous in our three-dimensional world. By contrast
most conventional transitions do not occur in infinite uniform two-dimensional
systems because of the increased role of thermal fluctuations. Here we explore
the dimensional crossover of Bose-Einstein condensation (BEC) for a weakly
interacting atomic gas confined in a novel quasi-two-dimensional geometry, with
a flat in-plane trap bottom. We detect the onset of an extended phase
coherence, using velocity distribution measurements and matter-wave
interferometry. We relate this coherence to the transverse condensation
phenomenon, in which a significant fraction of atoms accumulate in the ground
state of the motion perpendicular to the atom plane. We also investigate the
dynamical aspects of the transition through the detection of topological
defects that are nucleated in a quench cooling of the gas, and we compare our
results to the predictions of the Kibble-Zurek theory for the conventional BEC
second-order phase transition.
