Assume $M$ to be a compact $n$-dimensional manifold, endowed with a complete metric.

Let us consider the space $C^\infty(M)$ endowed with the standard $C^\infty$ topology, i.e. generated by the seminorms $$\sup_K\left|\frac{\partial}{\partial_x^\alpha}f\right|,$$ over the compact subsets of $M$. Now consider the following set: let $1\in C^\infty(M)$ be the constant function and $0<\varepsilon<<1$. Define $$A\doteq\left\{a\in C^\infty(M)\;\colon\;\|a-1\|_{C^1(M)}<\varepsilon\right\}\subset C^\infty(M).$$

It seems to me that Ascoli Arzelà applies here, in particular $A$ is relatively compact in $C(M)$.

The question is as follows:

is it possible to choose an open covering of $A$ such that all the balls are centered at smooth functions? The consequence of Ascoli Arzelà is that there exists these balls but in principle they may be centered just on continuous functions, not even differentiable, I was thinking about this stronger result because of the compactness of $M$ and (maybe) something along the lines of Stone-Weierstrass which allows to slightly modify the center of the balls of the covering so that they are indeed smooth functions. Is this possible?

Thanks for the attention.

smallballs, surely: as stated, for any $f \in A$, the (sup norm) ball centered at $f$ of radius $2 \epsilon$ contains all of $A$. $\endgroup$ – Nate Eldredge Aug 28 '14 at 14:48