  We report the observation of the Berezinskii-Kosterlitz-Thouless transition
for a 2D gas of exciton-polaritons, and through the joint measurement of the
first-order coherence both in space and time we bring compelling evidence of a
thermodynamic equilibrium phase transition in an otherwise open
driven/dissipative system. This is made possible thanks to long polariton
lifetimes in high-quality samples with small disorder and in a reservoir-free
region far away from the excitation spot, that allow topological ordering to
prevail. The observed quasi-ordered phase, characteristic for an equilibrium 2D
bosonic gas, with a decay of coherence in both spatial and temporal domains
with the same algebraic exponent, is reproduced with numerical solutions of
stochastic dynamics, proving that the mechanism of pairing of the topological
defects (vortices) is responsible for the transition to the algebraic order.
Finally, measurements in the weak-coupling regime confirm that polariton
condensates are fundamentally different from photon lasers and constitute
genuine quantum degenerate macroscopic states.
