Given a smooth vector bundle $E$ with non-compact base, let $\Gamma(E)$ be the space of $C^\infty$ sections equipped with compact-open $C^\infty$-topology.
I have heard that $\Gamma(E)$ is not locally-contractible. Why not?
Is $\Gamma(E)$ contractible? Visibly any section can be joined to the zero section by "straight line", doesn't this prove that $\Gamma(E)$ is contractible?
Is it true that every convex subset of $\Gamma(E)$ is contractible? The argument of 2 seems to apply, but then it seems plausible that each section has an arbitrary small convex neighborhood, contradicting 1.
CLARIFICATION: One source of "rumor 1" is the book "The Convenient Setting of Global Analysis" freely available here. On page 429 one reads: "Unfortunately, for non-compact $M$, the space $C^\infty(M, N)$ is not locally contractible in the compact-open $C^\infty$-topology". Another source is the discussion in Hirsch's book in the beginning of Chapter 2, which says "It can be shown that $C^\infty(M, N)$ has very nice features, e.g. it has a complete metric, and a countable base; if $M$ is compact, it is locally contractible and $C^\infty(M, \mathbb R^n)$ is a Banach space."
Thus I assumeed that in general, if $M$ is non-compact, then the space $\Gamma(E)$ is not (or maybe just need not be?) locally contractible. Also I am uncertain whether $C^\infty(M, \mathbb R^n)$ or $\Gamma(E)$ is a topological vector space, is it really? There seems to be a sequence of semi-norms giving these spaces a structure of Frechet spaces, but then they must be locally convex, hence locally contractible. Obviously, I am missing something.