I wondering how can I build a homeomorphism between two squares and a ellipsoid.

Question

For example, if we have X equal the union on the squares $[0,1] \mbox{x}[0,1]\mbox{x}{0}$ and $[0,1] \mbox{x}[0,1]\mbox{x}{1}$ and $Y= (x^2)/4 + (y^2)/4 + (z^2)/1 = 1$. How can I construct a homeomorphism between $X$ and $Y$? (I think after find this function we should use the theorem which says: " Let $X$ and $Y$ be topological spaces and let $f:X→Y$. If $X$ is compact, $Y$ is Hausdorff, and $f$ is a continuous bijection, then $f$ is a homeomorphism between $X$ and $Y$." to prove that it is a homeomorphism).

Answer 1

They are not homeomorphice, since $([0,1]\times[0,1]\times\{0\})\cup([0,1]\times[0,1]\times\{1\})$ is disconnected, whereas the ellipsoid is connected.