Prove the surface $\mathbb{S}^2\subset\mathbb{R}^3$ is homeomorphic to $\{(u,v)\in\mathbb{R}^2:u^2+v^2\le 1\}$ where $\mathbb{S}^2=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=1\}$. You may assume $\mathbb{S}^2$ is equipped with the subspace topology inherited from the standard topology on $\mathbb{R}^3$.
So my understanding is that we are done if we can find a homeomorphism from one to the other (i.e. a function which is bijective and bicontinuous). However the problem I'm having with this is firstly that I cannot think of such a function. Secondly, what is the point of the specific topologies here? I cannot seem to figure out why the specific topology assigned should make a difference.