Let $\sim$ relation be on $[0,1]$ as following.
$ x \sim y \Leftrightarrow x=y$ or $x,y \in \{0,1\}$
Show that $[0,1] / \sim$ is homeomorphic to $S=\{x\in \mathbb R^2 : \|x\|=1 \} \subseteq \mathbb R^2$
I need to concrete this quotient set’s elements. What are equivalence classes and what do this set’s elements look like? Also I am not sure on whether I have shown that it is an equivalence relation truly. Showing only one of three conditions of equivalence relation will provide me verifying myself.
It is easy I know but I am confused on this. Thanks for any help
Best Answer
For the homeomorphism part, identify $\mathbb{R}^2$ with the complex plane.
A natural choice for a homeomorphism is $f:[x]\mapsto e^{2\pi ix}$. This function is surjective because the complex logarithm inverts it on it's image. The injectivity follows from $f([x])=f([y]) \Rightarrow x-y\in\mathbb{Z} \Leftrightarrow x,y \in \left\{0,1\right\}\Leftrightarrow [x]=[y]$.
$f$ is continuous, because the complex exponential function is holomorphic. The continuity of $f^{-1}$ follows from the complex inverse function theorem.
With kindest regards,
soucerer