Invariance of domain and dimension of manifolds

Suppose $U$ is an open set about $0 \in [0,\infty ) = H^{1}$ and $V$ is an open set in $\mathbb {R}$. Show that $U$ is not homeomorphic to $V$.

Theorem (Invariance of domain): Suppose $U$ is an open subset of $\mathbb {R}^{n}$ and $f: U \to \mathbb {R}^{n}$ is $1$--$1$ and continuous. Then $f$ is an open map; that is, it maps open sets to open sets.

Apply this theorem to prove the following version for $n$-manifolds: if $M^{n}, N^{n}$ are $n$-manifolds and $f: M^{n} \to N^{n}$ is $1$--$1$ and continuous, then $f$ is an open map. Show that if we also assume that $M^{n}$ is compact, connected and $N^{n}$ is connected, then $f$ must be a homeomorphism.

Deduce the fact that the dimension of an $n$-manifold is well defined from the invariance of domain.

Prove that the invariance of domain holds when $n = 1$.

Show that a compact $n$-manifold has only a finite number of path components, each of which is a compact, connected $n$-manifold. Conclude that a compact $1$-manifold is the disjoint union of a finite number of circles.

Show that a $2$-sphere $S$ cannot be embedded in the plane.