When are two vector fields $C^1$ close? For simplicity let us assume we have a fixed compact manifold M. 
Introduction:
While considering $C^\infty(M,\mathbb{R})$ one can say that two functions $f,g$ are $\epsilon-close$ iff for fixed finite cover of $M$ by maps and compact sets contained in them $(\phi_i,U_i,K_i)$ we have $|f-g| \le \epsilon$ on every $K_i$ and their partial derivatives with respect to that maps lso satisfie $|\frac{\partial f}{\partial \phi_j}-\frac{\partial g}{\partial \phi_j}| \le \epsilon$ on every $K_i$. 
Question: I am reading article in which authors uses freely the notion of two vector fields on $M$ to be $C^1$ close without claryfing that notion. Is there some place where I can find definition of that topology? I have looked in google carefuly yet didn't find anything. 
 A: There are multiple ways of doing this; you want to write down a norm on the space of vector fields, and then say that two vector fields are close if they are in a sufficiently small neighborhood of each other in the topology this generates. You may get different norms here, but they'll be equivalent.
First, your manifold needs a Riemannian metric before you can possibly compare how big two tangent vectors are. After that, there are two really different ways of defining the norm.
1) Pick a finite open cover of $M$ by charts $U_n$ and a partition of unity $\rho_n$ subordinate to it. These charts come equipped with a trivialization of the tangent bundle over each chart (by taking the basis at a point $p$ to be the $\partial/\partial x_i$. The tangent bundle above each point still does have a Riemannian metric, probably different from the Euclidean metric. Then the norm is given by $\|X\|_{C^1} = \sum_n \|\rho_n X\|_{C^1}$, where I'm defining this norm on each chart in the obvious way: $\rho_n X$ is just a compactly supported function $\Bbb R^n \to \Bbb R^n$, so I just take the $C^1$ norm. Here the appearance of charts is to be able to take the derivative at all. (You could also define Sobolev norms this same way or take multiple derivatives to get $C^k$ norms.)
2) Note that vector fields still have an automatic notion of $C^0$-closeness coming from the metric; indeed, so do things in any tensor bundle (those also automatically get a metric). So define $\|X\|_{C^1} = \|X\|_{C^0} + \|\nabla X\|_{C^0}$, where $\nabla$ is the Levi-Civita connection, so that $\nabla X$ is a section of $TM \otimes T^*M$. Again, you could define Sobolev norms this way, and it's also clear how to iterate this construction to get $C^k$-norms.
As a small note on other vector bundles: if you want to define a $C^0$ norm on sections of a vector bundle $E$, all you need is a metric on that bundle $E$. But if you want to define $C^k$ norms you want a connection on the maniold, so that you can write $\|\nabla \sigma\|_{C^0}$, where now $\nabla \sigma$ is a section of $E \otimes T^* M$.
You may also be dispeased about the notion of "close" instead of a specific amount of closeness. The point is that in practice it's not really reasonable to write down precisely how close you need to be, just that you're close enough. As an example at a lower level: I have a space $X$ and a function $f$ with $f(p)=1$. "For points sufficiently close to $p$, $f(x) \neq 0$." The notion of closeness is to be able to say things like this - no need to actually say precisely how close in any way (especially because the various things I defined above are equivalent but not the same).
