Find a subspace homeomorphic to the quotient.

Question

I'm giving the equivalence relation $V\sim W$ when $\exists\lambda>0:\lambda V=W$. I have to prove that $\sim$ is an equivalence relation and find a subspace of $\mathbb{R}^2$ homeomorphic to the quotient space.

Here's what I did

$(1)$ $\sim$ is an equivalence relation

$(1.1)$ With $\lambda =1$ is $V\sim V$

$(1.2)$ If $V\sim W$ then$\lambda V=W$ which implies $\frac{1}{\lambda}W=V$ and then $W\sim V$

$(1.3)$ Suppose $V\sim W,W\sim Y$. If $\lambda_1 V\sim W$ and $\lambda_2 W\sim Y$ then $\lambda_1 \lambda_2 V \sim Y$ then $V\sim Y.$

$(2)$ Find a subspace homeomorphic to $(\mathbb{R}^2-\{(0,0)\})/\sim$.

I couldn't think of any example, hints for this part?

Answer 1

Hint:

Think of surjective and continuous function $f:\mathbb{R}^{2}-\left\{ \left(0,0\right)\right\} \rightarrow S^{1}$ prescribed by: $$v\mapsto\frac{v}{\left\Vert v\right\Vert }$$ and note that the equivalence classes are its nerves.