Nonnormality of \(\mathcal C^\infty (M,N)\) in Whitney's and related topologies when \(M\) is open
type: Item
instance of: scholarly article
title: Nonnormality of \(\mathcal C^\infty (M,N)\) in Whitney's and related topologies when \(M\) is open
zbMATH Open document ID: 0733.54013
DOI: 10.1016/0166-8641(91)90012-B
author: Vítor Neves
published in: Topology and its Applications
publication date: 1991-01-01T00:00:00Z
review text: The author answers the following question. Is either the Whitney \(\mathcal C^\infty\)-topology or Michor's extension of the Schwartz \(\mathcal D\)-topology on \(\mathcal C^\infty(M,N)\) paracompact, when \(M\) is an open manifold? What the author shows is that \((\mathcal C^\infty(M,N),\tau)\) is not even normal for any topology \(\tau\) finer than Whitney's and coarser than \(\mathcal D\). The author shows this by embedding Van Douwen's nonnormal space into a closed subset of \(\mathcal C^\infty(M,N)\). The author uses nonstandard analysis and well-known nonstandard characterizations to establish this major result.
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 54J05
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 58D99
Mathematics Subject Classification ID: 54A10
zbMATH DE Number: 4212980
zbMATH Keywords: Whitney topology
zbMATH Keywords: Schwartz topology
zbMATH Keywords: Michor topology
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4212980