One version of Whitney's approximation theorem states the following:

Let $N$ be a smooth, Hausdorff, second-countable (or paracompact) manifold, then given any continuous function $F:N\to \mathbb{R}^k$ and any continuous function $\epsilon:N\to\mathbb{R}_+$, one can approximate $F$ by a smooth function $\tilde{F}$ such that the error is bounded by $\epsilon$.

The proof I know uses the fact that (a) the statement is true for $N$ being an open subset of $\mathbb{R}^n$ (which one has Whitney's version with analytic approximation, or one can get some sort of $C^\infty$ approximation via mollifiers) (b) one can take a partition of unity on $N$.

**Question**

Are there analogues to Whitney's theorem when second-countable is weakened? Or is there an easy counterexample?