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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?

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    $\begingroup$ Shouldn't $X^{n+1}$ be based on $\mathbb{R}^{n+1}$ instead of $\mathbb{R}^n$? $\endgroup$ – Henno Brandsma Mar 5 at 10:10
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    $\begingroup$ 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 $\endgroup$ – Patrick Mar 5 at 10:13
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    $\begingroup$ 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. $\endgroup$ – Michael Burr Mar 5 at 10:16
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    $\begingroup$ 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? $\endgroup$ – stressed out Mar 5 at 10:16
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    $\begingroup$ 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? $\endgroup$ – Patrick Mar 5 at 10:22

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