Cornered sphere homemorphic to unit sphere A buddy of mine asked me this question, to which I found it intuitively obvious, but was unable to come up with a proper proof. Consider the so-called cornered sphere, defined by $x^4+y^4+z^4=1$ in contrast with the standard unit sphere $x^2+y^2+z^2=1$. It is pretty obvious when you make a picture that the cornered sphere is homeomorphic to the sphere proper. However, it seems hard to prove bceause the sort of natural map between the sets is not injective. That is to say, if you use the map $f(x,y,z) = (x^2, y^2, z^2)$ is not a homeomorphism. But surely an explicit map exists?

**Hint:** Try $f: \text{cornered sphere} \to \mathbb{S}^2(1)$ given by $$f(x,y,z) = (x,y,z) = \frac{1}{\sqrt{x^2+y^2+z^2}}(x,y,z).$$ This map projects the cornered sphere into the unit sphere, radially.