If $\textbf{x} \sim \textbf{y}$ iff $\textbf{x}=\lambda \textbf{y}$ for some $\lambda \in \mathbb{R}-\{0\}$.
If $\textbf{x} \sim_+ \textbf{y}$ iff $\textbf{x}=\lambda \textbf{y}$ for some $\lambda \gt 0$.
Let $X^{n+1}=\mathbb{R}^n-\{0\}$ in the standard topology for $\mathbb{R}^n$.
Show $X^{n+1}/\sim_+$ is homeomorphic to $S^n$
Show $X^{n+1}/\sim$ is too a quotient space of $S^n$. What are the projection map's equivalence classes?
- Find an imbedding of $X^{n+1}/\sim$ in $X^{n+2}/\sim$
I really just don't understand this material very well, so any hint would be really appreciated. Context: My first thought to solve question 1 is to say that the equivalence classes of $X^{+1}/∼_+$ are represented by the unit vectors in every direction from the origin. Hence, projecting each equivalence class onto the unit sphere would be a homeomorphism and likewise taking each point on the unit sphere to the ray from (0 to infinity) in that direction would suffice. Am I on the right track?