Many function-spaces are not normal if the domain is not compact
\textit{V. Neves} [Topology Appl. 39, No. 2, 113--122 (1991; Zbl 0733.54013)] has proved that \(C^\infty(M,N)\) with Whitney's \(C^\infty\)-topology or Michor's extension of Schwarz's \(\mathcal D\)-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. The author recovers 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 \textit{M. W. Hirsch} [Differential Topology. New York etc.: Springer-Verlag (1986; Zbl 0356.57001), p. 65].
scientific article; zbMATH DE number 4062126
scientific article