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

\(\displaystyle x\sim y\) 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

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