Many function-spaces are not normal if the domain is not compact

Abstract

V. Neves [4] has proved that C∞(M, N) with Whitney's C∞-topology or Michor's extension of Schwartz's D-topology is not a normal topological space provided that M is not compact. This result was shown by giving a closed embedding of Van Douwen's non-normal space using means of non-standard analysis. In this paper we recover this theorem by standard-techniques and by working in the function-space itself instead of giving an embedding. A similar method is used to obtain the same result for various other function-spaces in the case that the domain is not compact: spaces of continuous functions and C ^k-functions with Whitney's topology and spaces of sections of arbitrary differentiability-classes. Even any subspace of these spaces with non-empty interior is not normal, for example the spaces of immersions, embeddings, Riemannian metrics and symplectic structures. This also answers an open problem posed by Hirsch [2].