I want to prove that $\mathbb{R}^n$ contains no subspace homeomorphic to $S^n$, as per the wikipedia article
I'm trying to set up a function $g:S^n \to \mathbb{R}^n$ that would be injective, which cannot occur by Borsuk-Ulam.
If I have a subspace, $A \subset \mathbb{R}^n$, homeomorphic to $S^n$ then I can have a map $h:S^n \to A$. Can I compose this with the inclusion $i:\hookrightarrow \mathbb{R}^n$ to give an (injective) map $S^n \to \mathbb{R}^n$, and am done?
(As an aside, I am sure there are other point-set topology ways to prove this problem, maybe using invariance of domain)
Edit: Since the above works, to make this a little more interesting - can anyone provide any other proofs without using algebraic topology? (Although this is probably debatable, because some other methods to use it are probably based on algebraic topology)