# Quotient space homeomorphic to sphere.

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 $$\mathbb{R}^n$$.

1. Show $$X^{n+1}/\sim_+$$ is homeomorphic to $$S^n$$

2. Show $$X^{n+1}/\sim$$ is too a quotient space of $$S^n$$. What are the projection map's equivalence classes?

3. 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 $$X^{𝑛+1}/∼_+$$ 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?

• Shouldn't $X^{n+1}$ be based on $\mathbb{R}^{n+1}$ instead of $\mathbb{R}^n$? – Henno Brandsma Mar 5 at 10:10
• Thank you again for responding :). I think you're right. I've updated the question to correct that. My first thought to solve question 1 is to say that the equivalence classes of $X^{n+1}/\sim_+$ 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? I've – Patrick Mar 5 at 10:13
• Sure, there is a set-theoretic bijection between $X^{n+1}/\sim_+$ and $S^n$ (after changing $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$). Now you have to show that it is a homeomorphism, e.g., the map to $S^n$ is a quotient map. – Michael Burr Mar 5 at 10:16
• What have you tried so far? In the first one, no identification is taking place because $\lambda>0$. In the second one, antipodal points are identified. How can you make this argument rigorous? – stressed out Mar 5 at 10:16
• I'm really not sure how to make this rigorous. My first idea is to say let h be the homeomorphism that takes the equivalence class consisting of $\lambda x$ (for $||x||=1$ to the point $x\in S^n$. I have taken very few proof-based classes, so I'm terrible with making these notions rigorous. Is this on the right track? – Patrick Mar 5 at 10:22