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I understand some basic examples of homeomorphisms such as Show that $\mathbb{R}^2/{\sim}$ is homeomorphic to the sphere $S^2$. but I don't know where to start with this example. Do I have to consider/construct a quotient map in order to define a homeomorphism? Looking for some hints/starts.

$D^n$ is the disc $D^n = \{ x \in \mathbb{R}^n : |x| \leq 1 \}$

Edit: Still looking for help on this.

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grayQuant
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    What is $D^n$, and what is the basis of it? Up to homeomorphism, probably the closed $n$-ball and its boundary, but can you give the definitions you use? – Daniel Fischer Sep 28 '15 at 21:00
  • @DanielFischer Edited to add the definitions, thanks. – grayQuant Sep 28 '15 at 21:55
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    Can you map $D^n$ homeomorphically to a closed hemisphere of $S^n$? – Daniel Fischer Sep 28 '15 at 21:57
  • Yes that is possible, but I can't think of an explicit mapping – grayQuant Sep 28 '15 at 22:04
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    You can fairly easily find an explicit map if you consider $D^n$ as a subset of $\mathbb{R}^{n+1}$, $D^n = {(x_1,\dotsc,x_n,0) : x_1^2 + \dotsc + x_n^2 \leqslant 1}$, and $S^n$ as the unit sphere in $\mathbb{R}^{n+1}$. – Daniel Fischer Sep 28 '15 at 22:06
  • Note that if you construct a continuous bijection $D^n\times {-1,1}/{\sim}\to S^n$, it will automatically be a homeomorphism since the domain is compact and the codomain is Hausdorff. This lets you avoid having to muck around with exactly what the quotient topology looks like. – Eric Wofsey Sep 29 '15 at 05:58

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