Exercises

1. Prove the second equality in equation (41.1).
2. Prove corollary (41.6).
3. Prove that there is no injective continuous mapping from $S^n$ into $\mathbb{R}^n$. ([11], p. 217)
4. Show that no proper subset of $S^n$ can be homeomorphic to $S^n$. ([11], p. 217)
5. Let $\Omega$ be an open subset of $\mathbb{R}^n$ and be an injective continuous map. Show that $f$ is a homeomorphism onto its image. ([11], p. 217)