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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?

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  • $\begingroup$ try Counterexamples in Topology (Dover reprint of 1978 ed.) $\endgroup$
    – Nikita Kalinin
    Commented Dec 15, 2011 at 20:43
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    $\begingroup$ @Nikita: is that a general comment or do you know a particular one of the entries in that book would work? Checking the index I didn't see anything for "Whitney", "Approximation" or "Partition of Unity". The entries for "second countable" didn't really illuminate me. I have the Dover print of Counterexamples, so if you can just give me a rough page number it'd be great. (As a side note, I didn't think that anything in that book actually addresses smooth structures...) $\endgroup$
    – Willie Wong
    Commented Dec 16, 2011 at 9:37

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