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.

1. I have heard that $\Gamma(E)$ is not locally-contractible. Why not?

2. 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?

3. 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.