Defining an infinitely long cylinder.

Question

Question Define an infinitely long cylinder.

Definition: Quotient topology

Let $\left ( X,\tau \right )$ be a topological space. Let ~ be an equivalence relation on X and let $X^{*}$ be he collection of equivalence classes. Let $\pi: X\rightarrow X^{*}$ be the map taking each point to its equivalence class, i.e. $\pi\left ( x \right )=\left [ x \right ]$.

The quotient topology $\tau^{*}$ on $X^{*}$ is the collection of sets $U \subseteq X^{*}$ such that $\pi^{-1}\left ( U \right ) \in \tau$.

I am having a bit of problem beginning this question. A cylinder has a disc at the end. So perhaps there should be an $S^{1}=\left [ 0,1 \right ]$ we could work from?

However, any useful hints to kickstart me would be utmost helpful.

Thanks in advance.

Answer 1

Your definition of "cylinder" is incorrect: a cylinder does not have "caps" at the end. It should just be the "tube" part. (As a side note, you shouldn't write e.g. "$S^1=[0, 1]$" - "$S^1$" is the standard notation for the $1$-sphere, that is, the unit circle. In your previous edit, what you described as "$1$-sphere" is actually called a "$2$-disc" or "$2$-ball" - an $n$-sphere is the boundary of a $(n+1)$-ball. The "$n$" refers to the dimension of the object - so a circle is one-dimensional, even though it "lives" most naturally in two-dimensional space, and the cap at the end of a cylinder is two-dimensional.)

Intuitively, you can form an infinite cylinder by taking $\mathbb{R}^2$ and "rolling it up". The point of the question is to make this precise by formalizing it as a quotient space construction. HINT: let's say I roll the plane along the $x$-axis, and wind up with a cylinder with circumference $1$. Note that the points $(0, 0)$ and $(1, 0)$ wind up "overlapping." Do you see, in general, which points overlap with which other points?