Many function-spaces are not normal if the domain is not compact
✍ Scribed by K. Wegenkittl
- Publisher
- Springer
- Year
- 1989
- Tongue
- English
- Weight
- 346 KB
- Volume
- 7
- Category
- Article
- ISSN
- 0232-704X
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✦ Synopsis
V. Neves [4] has proved that C (M,N) with Whitney's C-topology or Michor's extension of Schwartz's -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-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].