Let $\displaystyle X=\mathbb{R}$ then define an equivalence relation $\displaystyle \sim$ on $\displaystyle X$ s.t.

$\displaystyle x\sim y$ if and only if $\displaystyle x-y\in\mathbb{Z}$

Show that $\displaystyle X/\sim\cong S^1$

So denoting the elements of $\displaystyle X/\sim$ as $\displaystyle [t]$

The function

$\displaystyle f([t])=\exp^{2\pi ti}$ defines a homemorphism.

$\displaystyle f([t])$ is continuous since $\displaystyle \exp^{2\pi ti}=cos(2\pi t)+isin(2\pi t)$ which is the sum of continous functions.

Letting $\displaystyle f([x])=f([y])\Rightarrow\exp^{2\pi xi}=\exp^{2\pi yi}\Rightarrow2\pi xi=2\pi yi\Rightarrow x=y$ so injective.

Now $\displaystyle f([t])=\exp^{2\pi ti}=z$ for $\displaystyle z\in\mathbb{C}$ s.t. $\displaystyle |z|=1$

Then $\displaystyle t=\frac{-i}{2\pi}log(z)=\frac{-i}{2\pi}(log|z|+iArg(z))$

Since $\displaystyle |z|=1$ we have $\displaystyle log|z|=0$ so $\displaystyle t=\frac{1}{2\pi}Arg(z)$ which is in $\displaystyle X/\sim$ so surjective.

Therefore a bijection.

Not sure how to show $\displaystyle f^{-1}$ continuous?

Is this correct? Any input would be great. Thanks