# Local compactness of $C^k(M,N)$ strong space.

Let $M$ and $N$ be two $C^k$ manifolds with $k\geq 1$, with $M$ non compact. I know that $C^k(M,N)$ with its strong (Whitney) topology isn't metrizable and that it's a Baire space. Can I prove that it's a locally compact and/or Hausdorff space or do you have any counterexample ? Thank you!

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Can you handle the case of a simple example, like $M = N = \mathbb{R}$? – t.b. Sep 12 '12 at 12:38

The space is Hausdorff because if $f$ and $g$ differ at some point, they can be separated by $C^0$ neighborhoods which are larger than $C^k$ neighborhoods.
But not locally compact, not even sequentially. The issue is that the uniform bound on the $k$-th derivative of $f_n$ does not imply the existence of the $k$-th derivative at the limit. Consider the example $f_n(x)=(x^2+1/n)^{1/2}x^{k-1}$.
Hello, I indeed found the Hausdorff parts of my question, even though I must admit I had a few problems writing it in all generality (choosing the right charts in $N$). I was stuck with local compactness, because I'm not very used using these topologies. Using sequences seems useful in this case. Thank you. – Philippe Malot Sep 14 '12 at 20:17