$(S^1 \times [0,1])/\sim$ homeomorphic to unit disk $ D^2$
Let $\sim$ be the equivalence relation on $S_1 \times I$ given by $(x,t)\sim(y,s)$ if and only if $xt=ys$ where $I=[0,1]$
How do I prove that $(S_1 \times I)/\sim$ is homeomorphic to the unit disk $D^2$ with the induced topology?
Solution 1:
The map $f:S^1×I→D^2$ which maps $(x,s)$ to $xs$ is continuous, closed, and surjective as each $y\ne 0\in D^2$ is the image of $\left(\frac y{||y||},||y||\right)$. Since $f(x,s)=f(y,t)\iff xs=yt\iff (x,s)\sim (y,t)$, the universal property of quotient spaces induces a homeomorphism $\dfrac{S^1×I}\sim \to D^2$