Many function-spaces are not normal if the domain is not compact
type: Item
instance of: scholarly article
title: Many function-spaces are not normal if the domain is not compact
zbMATH Open document ID: 0651.54005
DOI: 10.1007/BF00128297
author: K. Wegenkittl
published in: Annals of Global Analysis and Geometry
publication date: 1989-01-01T00:00:00Z
review text: \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].
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 4062126
zbMATH Keywords: normal topological space
zbMATH Keywords: Whitney topology
MaRDI profile type: Publication
cites work: Q4046802
cites work: Differential Topology
cites work: Q3871594
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description: scientific article; zbMATH DE number 4062126