Let $\mathbb{R^3}$ be given the standard topology. Let $P$ be a sextic parabloid and $H$ be the circular half-hyperboloid in $\mathbb{R^3}$ defined by
$P = {(x,y,z) ∈ \mathbb{R^3} : y=x^6 + z^6}$
$H = {(x,y,z) ∈ \mathbb{R^3} : z^2 -1= x^2 +y^2, z \geqslant 0}$
Consider $P$ and $H$ as topological spaces with the subspace topology. Prove that $P$ and $H$ are homeomorphic.
I am struggling with this question and don't really know where to begin. I know how to prove that elementary functions are homeomorphic but have no idea how to do this for $P$ and $H$. Any help will be much appreciated.
So far, I have:
For the function, $F:\mathbb{R^2} \rightarrow P$ given by $F(x,z) = (x, x^6 + z^6, z)$ is a homeomorphism since it is continous and bijective, and its inverse, $F^{-1}:P \rightarrow \mathbb{R^2}$, given by $F^{-1}(x,x^6 + z^6,z)=(x,z)$ is also continous. I've done the same thing for the function G for the space H, but don't know where to go from here.
Let me answer with a hint.
The equation $y = x^6 + z^6$ is the graph of a function $y=f(x,z)$ which is clearly defined defined for all $(x,z) \in \mathbb R^2$.
Similarly, the equation $z = \sqrt{x^2 + y^2 + 1}$ is again the graph of a function $z=g(x,y)$, again defined for all $(x,y) \in \mathbb R^2$.
Visualizing those graphs, what familiar topological space are they both homeomorphic to?
For a bigger hint: Consider the functions $$F(x,z) = (x,f(x,z),z) $$ and $$G(x,y) = (x,y,g(x,y)) $$ Describe the domain and image of each. Prove that each is continuous and one-to-one. Then prove that each of their inverse functions, from their image to their domain, is continuous.