I am reading Audin and Damian's book on Morse Theory and Floer Homology. In pages 40-41 they state the Smale Theorem, whose statement uses a notion of $C^1$-proximity for vector fields.
I believe the definition of proximity herein described has an issue.
I think the order of the quantifiers should be: "for every $\epsilon > 0$, there exists a vector field $X'$such that for every cover of V by charts..."
The way it is written, I could conceive of vector fields $X$ and $X'$ such that in an open chart $\varphi(\mathbb{B}^n)\subset V$ the first is $$T\varphi^{-1}(X)(p) = a_1(p)\frac{\partial}{\partial x_1}\vert_p+\cdots+a_n(p)\frac{\partial}{\partial x_n}\vert_p$$ and the second is $$T\varphi^{-1}(X')(p) = b_1(p)\frac{\partial}{\partial x_1}\vert_p+\cdots+b_n(p)\frac{\partial}{\partial x_n}\vert_p.$$
And they satisfy $$\lVert T\varphi^{-1}(X')(p) - T\varphi^{-1}(X)(p)\rVert<\epsilon$$ on compacts subsets of the unit ball.
Then if we set now $\tilde{\varphi}:2\mathbb{B}^n\to \varphi(\mathbb{B}^n)\subset V$, that is, we parametrize $\varphi(\mathbb{B}^n)$ by the ball of radius 2, and $\tilde{\varphi}^{-1}_i(p) = 2\varphi^{-1}_i(p)$, we end up with
$$T\tilde{\varphi}^{-1}(X)(p) = 2a_1(p)\frac{\partial}{\partial \tilde{x}_1}\vert_p+\cdots+2a_n(p)\frac{\partial}{\partial \tilde{x}_n}\vert_p$$ and the second is $$T\tilde{\varphi}^{-1}(X')(p) = 2b_1(p)\frac{\partial}{\partial \tilde{x}_1}\vert_p+\cdots+2b_n(p)\frac{\partial}{\partial \tilde{x}_n}\vert_p,$$
so that $$\lVert T\tilde{\varphi}^{-1}(X')(p) - T\tilde{\varphi}^{-1}(X)(p)\rVert<\epsilon$$ may be false on certain compacts, since for the same $p$ it is double the original $\lVert T\varphi^{-1}(X')(p) - T\varphi^{-1}(X)(p)\rVert<\epsilon$ (we may take the compact $K$ first to be say the closed ball of radius $\frac{1}{2}$, and then the closed ball of radius $1$ for the computation of the norm). This would mean that by changing the compact or open cover chosen we break the $\epsilon$ bound. That is, this notion of $C^1$ proximity doesn't seem to depend solely on the $X$ and $X'$.
Would someone be able to clarify what this notion of closeness is?
The authors do not seem to state that $V$ is compact but I assume that is intended.
I think the order of quantifiers is not correct as stated (but you can read it in a way so that it makes sense), but yours is not what they are asking for either. I will explain the conceptual content of their statement, from which we can recover the intended order of quantifiers.
Audin and Damien are introducing a norm on the space $\mathfrak X(V)$ of vector fields on $V$: they want to say "There is a vector field $X'$ satisfying the desired properties, which is close to $X$ with respect to this norm."
The construction that Audin and Damien have in mind is the following. Given a chart $\varphi: \Bbb R^n \to V$, one may pull back the vector field to $\Bbb R^n$. Given a compact set $K \subset \Bbb R^n$, we may then measure the $C^1$-norm of $X$ with respect to $\varphi$ and $K$ as $$\|X\|_{C^1, \varphi, K} = \sup_{p \in K} \left(\|d\varphi^{-1}(X_{\varphi(p)})\| + \|\nabla d\varphi^{-1}(X_{\varphi(p)})\|\right),$$ where we take the gradient by thinking of $d\varphi^{-1}(X)$ as a map $\Bbb R^n \to \Bbb R^n$.
Now given a finite collection $\mathcal C$, whose elements are $(\varphi_i, K_i)_{1 \leq i \leq n}$ pairs of charts $\varphi_i$ and compact sets $K_i \subset \Bbb R^n$, so that $\bigcup_i \varphi_i(\text{int}(K_i)) = V$, we may set $$\|X\|_{C^1, \mathcal C} = \max_{i=1}^n \|X\|_{C^1, \varphi_i, K_i}.$$
This is a norm on the space of vector fields. (It is a complete norm on the space of $C^1$ vector fields.) This norm depends on the choice of $\mathcal C$, but different choices give rise to \emph{equivalent} norms: if $N_1$ and $N_2$ are two norms arising from this process there is some $C > 0$ so that $N_1(X)/C \leq N_2(X) \leq C N_1(X)$.
Audin and Damien's claim is supposed to be: fix ahead of time a collection $\mathcal C$ as above. Then for any $\epsilon > 0$, there is a Morse-Smale pseudogradient field $X'$ with $\|X' - X\|_{C^1, \mathcal C} < \epsilon$.
This inequality will change wildly as we change $\mathcal C$, so we need to make such a choice ahead of time.
Thus the correct way to state their claim is: "For every cover of $V$ by compact subsets of charts, there is a vector field $X'$ so that (stated inequality) holds."
That is: you don't choose $X'$ after you choose a cover by charts and a specific compact subset of each chart so that $X'$ is small on that single compact subset. Rather, you choose a cover of charts, a compact subset of each chart, and then you want $X'$ to be small on all of those chosen compact subsets.
Different compact subsets will lead to a different notion of smallness. You will not be able to choose $X'$ so that it is $\epsilon$-close to $X$ with respect to all of these different notions of smallness.