Nonnormality of \(\mathcal C^\infty (M,N)\) in Whitney's and related topologies when \(M\) is open (Q809395)

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.