  We study the space of complete Riemannian metrics of nonnegative curvature on
the plane equipped with the C^k topology. If k is infinite, we show that the
space is homeomorphic to the separable Hilbert space. For any k we prove that
the space cannot be made disconnected by removing a finite dimensional subset.
A similar result holds for the associated moduli space. The proof combines
properties of subharmonic functions with results of infinite dimensional
topology and dimension theory. A key step is a characterization of the
conformal factors that make the standard Euclidean metric on the plane into a
complete metric of nonnegative sectional curvature.
