Let $\displaystyle \phi : \mathbb{R}^2 \rightarrow \mathbb{R}^3$, so:

$\displaystyle (u,v) \rightarrow (u^3,u+v^3,v)$

Show this is a homeomorphism:

To show injectivity: We want to show that:

$\displaystyle \phi(u_1,v_1)=\phi(u_2,v_2) \rightarrow (u_1,v_1)= (u_2,v_2) $

So if we set:

$\displaystyle u_{1}^{3}&=&u_{2}^{3}$

$\displaystyle u_1+v_1^3=u_2+v_2^3$

$\displaystyle v_1=v_2$

Then from the third equation we get $\displaystyle v_1=v_2$, and can sub back in to find that $\displaystyle u_1=u_2$.

The surjectivity is where I'm getting stuck. If we let any $\displaystyle (x,y,z)$ be a point in $\displaystyle \mathbb{R}^3$

Then we get $\displaystyle v=z$ and subbing in to the equation for y means that we get $\displaystyle u = y-z^3$.

So I guess I go on to say that given any $\displaystyle (x,y,z)$ I can find a $\displaystyle (u,v)$ such that $\displaystyle \phi(u,v)=(x,y,z)$

However, I think this is wrong since what happens with the $\displaystyle x$ value in this case? I mean we also get that $\displaystyle u=x^{\frac{1}{3}}$

I'm lost here can someone help? I don't think what I did initially was enough?

Thanks!