Suppose $M$ is a 3-dimensional manifold, John W. Morgan and Frederick Tsz-Ho Fong in their "Ricci Flow and Geometrization of 3-Manifolds" book as a definition of canonical neighborhoods have written:
We fix an $\epsilon >0$, there are essentially $3$ types of $\epsilon$-canonical neighborhoods:
(1) ($\epsilon$-neck) - a neighborhood $N_{\epsilon}\subset M$ diffeomorphic to $S^2\times (-\epsilon ^{-1},\epsilon ^{-1})$ under diffeomorphism $\varphi:S^2\times (-\epsilon ^{-1},\epsilon ^{-1})\to N_\epsilon$, such that the rescaled pull-back metric $R(x,t)\varphi ^*g(t)$ on $S^2\times (-\epsilon ^{-1},\epsilon ^{-1})$ is within $\epsilon$ in $C^{[\epsilon^{-1}]}$-topology to the product of the round metric on $S^2$ with $R=1$ with the usual metric on $(-\epsilon ^{-1},\epsilon ^{-1})$.
(2) ($\epsilon$-cap) - topologically $B^3$ or a punctured real projective $3$-space $\mathbb{R}P^3_0$ and whose end is a $\epsilon$-neck.
(3) connected component of positive sectional curvature.
A point $x\in M$ is said to have an $\epsilon$-canonical neighborhood if it lies in the central two-sphere of an $\epsilon$-neck, lies in an $\epsilon$-cap in the complement of the $\epsilon$-neck forming the end of the cap, or lies in a component of positive sectional curvature.
Question 1: What is the meaning of "within $\epsilon$ in $C^{[\epsilon^{-1}]}$-topology"?
Question 2: Why do we consider rescaled pull-back metric $R(x,t)\varphi ^*g(t)$? What is necessity of this action? In other words if we have just considered pull-back metric $\varphi ^*g(t)$, what would be wrong then?
Question 3: Is the boundary of the punctured real projective $3$-space ,$\mathbb{R}P^3_0$, $S^2$?
Question 4: I can visualize $\epsilon$-neck and $\epsilon$-cap. These types of canonical neighborhoods have shapes like following figures. Now, what is visualization of type (3) of the definition of the canonical neighborhoods? Essentially, what is "a component of positive sectional curvature" ?
Thanks in advance.
The answer to the question one:
They want to consider the semi-norms induced topology as a Fréchet Space, so basically is the topology of "almost" infinite differentiable functions (almost as epsilon is really small, so $\epsilon^{-1}$ is a really big number.
Question 2:
Is mostly used to be able to recuperate the action of the metric you crecovered in the original manifold, i mean, you have a metric which is defined on a component that is about to turn to a singularity and you want to study it in a different space, that is, the original manifold under the ricci flow action, thus, rescalling.
answer to question 4:
"a component of positive sectional curvature" is basically something that locally has positive sectional curvature, which turns out to be a sphere. (you can check DoCarmo´s book in riemannian geometry for details). Another apprach to visualizing this may be the spherical space forms (Wol´s book spaces of constant curvature has a quite etailed exposition in this matter) so it is a space where the geometry tells you to move according to a constant positive sectional curvature (in each point you "feel" like moving as you do in a sphere).
You also want to note that despite ricci flow uses, there is a normalized ricci flow which is volume-preserving and with it, you can study the changes in the topology and geometry of your manifold M without changing it "too much" (thus, perelman´s idea of recovering the original manifold uses the normalized ricci flow).