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 neighborhoodif 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.