On the existence of a continuous bijection from a quotient space to the unit sphere $S^2$

On the existence of a continuous bijection from a quotient space to the
unit sphere $S^2$

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.