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I'm trying to construct a quotient map $f$ from the $n$-sphere $S^n$ to the $n$-disk $D^n$, in which $f(x_1,x_2,\ldots,x_n,x_{n+1})=f(x_1,x_2,\ldots,x_n,-x_{n+1})$. The goal is to show that $S^n/\sim_f$ is homeomorphic to $D^n$. (Is this the right way to go about it?)

I've been looking at the stereographic projection for inspiration, but haven't come up with anything. I only just started learning about topology. Any hints?


marked as duplicate by Henno Brandsma general-topology Dec 11 '18 at 15:57

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  • $\begingroup$ You want to first define an equivalence relation on the sphere which it seems you have done. Then you need to define your bijection from the sphere (more specifically from Equivalence classes of sphere) to the disk. $\endgroup$ – user25959 Dec 11 '18 at 15:34
  • $\begingroup$ If you're willing to go the other way, you can get the homeomorphism directly: $x\mapsto (2\sqrt {1-\|x\|^2}x,2\|x\|^2-1)$ $\endgroup$ – Matematleta Dec 11 '18 at 16:14

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