If $X\approx Y$ and $X$ is convex then $Y$ might not be convex

Question

Let $X$ and $Y$ be subspaces of an euclidean space. If $X$ is homemorph to $Y$ that is $X\approx Y$ ($f$,$f^{-1}$ continuous and $f$ bijective) and $X$ convex. Prove that $Y$ is not necessary convex.

I found a counterexample in this link Are homeomorphisms convex-preserving?

$\varphi:(x,y)\mapsto (x,y^3)$. It is a homeomorphism ($\varphi^{-1}:(x,y)\mapsto (x,\sqrt[3]{y}$) and the image of the first diagonal is $y=x^3$ which is not convex.

I do not see clear what is $Y$ in this case.

Also what it means with "the image of the first diagonal is $y=x^3$" ?

where does it come from?

If someone could explain please, thank you.

Answer 1

Just for the sake of a proof without words: