Construct torus from the following cylinder Construct a torus as an identification space of the cylinder:
$S=\{(s,\sin t,\cos t)|0 \leq s \leq 2, 0 \leq t<2 \pi\}$
I know I need to "package up" points in $S$ into a collection of subsets $S*$ that classify the points in each subset as equivalent under the identification map $S \rightarrow S*$. Where $S*$ is the identification space of the torus.
So I think of partitioning points in $S$ into the following subsets:
Sets consisting of pairs of points of the form $(0,\sin t,\cos t),(2,\cos t, \sin t),0 \leq t<2 \pi$
Sets consisting of a single points $(s,\sin t, \cos t)$ where $0<s<2$, $0 \leq t< 2 \pi$
I'm new to topology, my nephew is studying it and told me about it, so bear with me.
 A: Yes, you are correct. It is similar to turning a line segment into a circle. In this case, we would have $[0, 2]$ on which we define the relation $\sim$ by: $x\sim y$ iff $x=y$ or $|x-y|=2$ (note that the latter condition can only be satisfied by the two end points). Then, the same idea for turning a cylinder into a torus. We have the relation $\sim$ on $S$ defined by $(s, \sin t, \cos t) \sim (s', \sin t', \cos t')$ iff $|s-s'|=0, 2$ and $t=t'$. This essentially divides first the cylinder into line segments from top to bottom that we then turn into circles, for any angle $t$. This would be a more precise formulation of what you wrote, but content-wise the same.
A: It seems that you know some topology, in particular the concept of an identification map (= quotient map).
Your space $S$ is nothing else than the product $[0,2] \times S^1$, where $S^1 = \{(x,y) \in \mathbb R^2 \mid x^2 + y^1 = 1 \} \subset \mathbb R^2$ is the standard unit circle in the plane. Let $p : [0,2] \to S^1, p(t) = (\cos (\pi t), \sin (\pi t))$. This is an an identification map which identifies the boundary points $0, 2$ of $[0,2]$ to a single point in $S^1$. Now define
$$q : S \to S^1 \times S^1, q(t,z) = (p(t),z). $$
This is again an identification map and the fibers $q^{-1}(v,w)$ with  $v,w \in S^1$ decompose $S$ into the sets you described in your question.
