Homeomorphism between region above parabola and $ \mathbb R ^2$

Question

Define the set $ X = \{ (x,y) \in \mathbb R ^2 : x^2 < y \}. $ Construct a homeomorphism between $ X $ and $ \mathbb R ^2 $.

Graphically, $ X $ is the region above the parabola $y = x^2$, not including the boundary lines. Since it's contained entirely in the upper half plane, I thought of trying something involving logarithms and considered $\Phi: X \to \mathbb R ^2, \; (x,y) \to (x, \ln \sqrt y)$. As it stands, $f$ is injective and also continuous as the composition of continuous maps on the domain of definition but I wasn't able to show surjectivity. For any $ (x,y) \in \mathbb R ^2 $ we have $(x,y) = \Phi (x, e^{2y} ) $, but the latter isn't necessarily contained in $X$ so I'm not sure how to proceed (or even if this kind of function is the right form to be considering).

Answer 1

You can do it in two steps:

1. Consider the map$$\begin{array}{rccc}\psi\colon&X&\longrightarrow&\mathbb{R}^2\\&(x,y)&\mapsto&(x,y-x^2).\end{array}$$Then $\psi$ is a homeomorphism from $X$ onto $\mathbb{R}\times(0,\infty)$.
2. Now, define a homeomorphism from $\mathbb{R}\times(0,\infty)$ onto $\mathbb{R}^2$ and compose both homeomorphisms.

Answer 2

Hint: For the strategy $(x,y)\mapsto(x,\star)$ to work, you'll want a function of both $x$ and $y$ in the $\star$.