  In this paper, we study open complete metric spaces with non-negative
curvature. Among other things, we establish an extension of Perelman's soul
theorem for possibly singular spaces: "Let X be a complete, non-compact, finite
dimensional Alexandrov space with non-negative curvature. Suppose that X has no
boundary and has positive curvature on a non-empty open subset. Then X must be
a contractible space". The proof of this result uses the detailed analysis of
concavity of distance functions and Busemann functions on singular spaces with
non-negative curvature. We will introduce a family of angular excess functions
to measure convexity and extrinsic curvature of convex hypersurfaces in
singular spaces. We also derive a new comparison for trapezoids in
non-negatively curved spaces, which led to desired convexity estimates for the
proof of our new soul theorem.
