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## Homeomorphism

General Topology
Papapetros Vaggelis
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### Homeomorphism

Consider the set $\displaystyle{X=S^{1}-\left\{\left(\cos\,\phi,\sin\,\phi\right)\right\}\subseteq \mathbb{R}^2}$ , where $\displaystyle{\phi\in\left(0,2\,\pi\right)}$.

Prove that there exists $\displaystyle{k\in\mathbb{Z}}$ such that the sets $\displaystyle{X}$ and $\displaystyle{\left(\phi+2\,k\,\pi,\phi+2\,(k+1)\,\pi\right)\subseteq \mathbb{R}}$

are homeomorphic.
Nikos Athanasiou
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### Re: Homeomorphism

Since the homeomorphism relation is an equivalence relation, suffices to show both spaces are homeomorphic to the open interval $(0,1)$.

The latter space certainly is.

For the first one, wlog $\phi = 0$ and parametrise $X$ as $\lbrace (\cos x , \sin x) \mid 0<x< 2 \pi \rbrace$ . Define $f : X \rightarrow (0,1)$ by

$$f((\cos x, \sin x)) =\frac{1} {2 \pi} \cdot x$$ .

This is a homeomorphism.

May I just say that any $k$ will do and in fact, any interval will do.
Papapetros Vaggelis
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### Re: Homeomorphism

Hi Nikos. Thank you for your solution.

I would like to make a geometrical comment.

My opinion is that $\displaystyle{k\in\mathbb{Z}}$ is the rotation index of the circle $\displaystyle{S^1}$

having the paramatrization $\displaystyle{S^1=\left\{\left(\cos\,x,\sin\,x\right)\in\mathbb{R}^2: x\in\mathbb{R}\right\}}$.

For example, if we want to paramatrize the circle from $\displaystyle{\left(\cos\,\phi,\sin\,\phi\right)}$ ($\displaystyle{k=0}$

then, $\displaystyle{X\simeq Y=\left(\phi,\phi+2\,\pi\right)}$ by defining $\displaystyle{f(x)=(\cos\,x,\sin\,x)\,,x\in Y}$ .

If we want to "run" the circle one time ($\displaystyle{k=1}$), then the above function with domain

$\displaystyle{\left(\phi+2\,\pi,\phi+4\,\pi\right)}$, is appropriate.