Diffeomorphism - Diffeomorphism Group

Let M be a differentiable manifold that is second-countable and Hausdorff. The diffeomorphism group of M is the group of all Cr diffeomorphisms of M to itself, and is denoted by Diffr(M) or Diff(M) when r is understood. This is a 'large' group, in the sense that it is not locally compact (provided M is not zero-dimensional).