Compactness of a Sphere

This fact is mentioned liberally in literature along with subsequent mention of Heine-Borel and I am trying to get my head around it. What would be a formal proof of this if we take for example, a unit sphere in $\mathbb R^3 $?

In addition, how could it be shown, non-trivially of course, that this sphere is not homeomorphic with $\mathbb R^2 $? As I understand this involves finding a bijective function with continuous inverse?

As yoyo says in the comments, you can use the compactness criterium that a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded.

So you can embed the sphere $S^2$ in $\mathbb{R}^3$ via the equation $x^2+y^2+z^2=1$. Satisfying an algebraic eqution is "clearly" a closed condition, so $S^2$ is closed (why is this clear? If you perturb the variables with arbitrarily small $\epsilon$, you will leave the sphere - this is what closedness means).

Why is it bounded? This is clear: the equation says precisely that any point on the sphere has absolute value $1$. So the sphere is clearly bounded.

It follows from Heine-Borel that the sphere is compact as a subset of $\mathbb{R}^3$. But compactness is a topological invariant, so we are done.