Is $C^k(X,Y)$ a Lindelöf space?

Let $X$ be a compact, finite dimensional smooth manifold and $k\in \mathbb{N}$. Consider the following two cases:

1. $Y$ is a finite dimensional manifold
2. $Y$ is an infinite dimensional Banach manifold

Question: Is the space $C^k(X,Y)$ a Lindelöf space in one or both of these cases?

Thanks for any insights. I also appreciate any references of results that go in this direction.

Edit: In case one assumes that $Y$ is second countable, I've come up with the following approach:

• From a comment of @Martin to this answer I gather that the space of $k$-jets $J^k(X,Y)$ is second countable if $X$ and $Y$ are second countable.
• A result (on page 2) in this paper implies that if $X$ and $Z$ are second countable, then $C^0(X,Z)$ (endowed with the compact-open topology) is separable.

Setting $Z:=J^r(X,Y)$ and using the fact that the $C^k$-topology on $C^k(X,Y)$ is induced by the $k$-jet extension $$j^k:C^k(X,Y)\to C^0(X,J^k(X,Y))$$ we can deduce that $C^k(X,Y)$ is separable. A metric space is separable if and only if it's Lindelöf. Since $C^k(X,Y)$ is a metric space we conclude that it is Lindelöf.

Does this approach work, or are there some problems that I'm overlooking?

If it does, some more details of the individual steps (especially why $J^k(X,Y)$ is second countable if $X$ and $Y$ are second countable) are also appreciated.

Thank you for any feedback.

• By your previous question you already know that the answer in case 1 is yes, as $C^k(X,Y)$ is a separable, completely metrizable space. – Martin Jun 6 '13 at 13:13
• @Martin: Thank you for pointing that out. – Dave Jun 6 '13 at 13:58