$C^2$ functions is an open subset of $C^0$?

Question

Let $C^k(\mathbb{R}^d;\mathbb{R}^D)$ denote the set of $k$-times differentiable functions from $\mathbb{R}^d$ to $\mathbb{R}^D$; where $d,D$ are positive integers. Is $C^k(\mathbb{R}^d;\mathbb{R}^D)$ an open subset of $C(\mathbb{R}^d;\mathbb{R}^D)$; when the latter is equipped with the compact-open topology (topology of uniform convergence on compacts)?

More generally, does it have a non-empty interior?

Answer 1

No linear subspace of a topological vector space $X$ other than $X$ itself can have any interior points. Proof: if $M$ is a linear subspace of $X$ with non-empty interior the we can use a translation to see that $0$ is an interior point of $M$. If $x \in X$ then $\frac x n \to 0$ so $\frac x n \in M$ for $n$ sufficiently large. But then $x \in M$. Thus $X=M$.