We ill derive an optimal extension of a comparison theorem of Perelman
for quadrangles in (possibly singular) spaces with non-negative curvature. Such a quadrangle comparison theorem can be viewed as a version of Berger Lemma for singular spaces.

      Among other applications of this new quadrangle comparison theorem, we ntend to establish the Cheeger-Gromoll-Perelman soul theory for singular spaces. For instance, we will show that ``if an open Alexandrov space M has on-negative curvature and has positive sectional curvature on a small open ball, then M must be contractible." This result extends the celebrated Perelman's soul theorem for singular spaces. This is a joint work with Bo Dai and Jiaqiang Mei.