There is a question from an old topology prelim that is somewhat giving me a hard time. Consider the cylinder $X= S^1 \times [-1,1]$. Now we define an equivalence relation $\sim$ as follows: For points $v,v' \in S^1$, we have $(v,-1) \sim (v',-1)$ and $(v,1) \sim (v',1)$. I am asked to show that the quotient space $X^{*}= S^1 \times [-1,1]/\sim$ is homeomorphic to the unit sphere $S^2$. The problem is I can't off the top of my head come up with a decent continuous bijection from the quotient space onto $S^2$. What might work here?
Suppose I had some sort of continuous bijection $h: X^{*} \rightarrow S^2$. Now the quotient map $p: X \rightarrow X^{*}$ is continuous and surjective, and since $X$ is compact, so is $X^{*}$. We also know that $S^2$ being a topological manifold is Hausdorff. Recall that if there is a continuous bijection between the compact space $X^{*}$ (any compact space for that matter) and the Hausdorff space $S^2$ (or any Hausdorff space), then that continuous bijection is a homeomorphism. This is what I intended to do, but I still can't come up with such a continuous bijection. Also, perhaps I am a bit confused in trying to visualize the quotient space. I would really appreciate some input on this, and any ideas that may prove useful.
Best Answer
Try to find for a point $((a,b),z)\in S^1\times [-1,1]$ a point $(x,y,z)$ in $S^2$ with the same $z$-coordinate and $x=\lambda a,\ y=\lambda b$ for a $\lambda$ which is a function in $z$.
Then show that the map $f:((a,b),z)\mapsto (x,y,z)$ is surjective and continuous, and that it induces a map $\tilde f:X^*\to S^2$. Is $\tilde f$ injective?