Homeomorphism math help needed

  • #1
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If [itex]D^n[/itex] is the unit n ball in Euclidean n-space. i.e.
[itex]D^n = \{ x \in \mathbb{R}^n : ||x|| \leq 1 \}[/itex]
and [itex]S^n[/itex] is an n-sphere.

how do i show that [itex]D^n / S^{n-1}[/itex] is homeomorphic to [itex]S^n[/itex]?
there's a hint suggesting i first of all try the n=1,2,3 cases. where [itex]X/Y= X \backslash Y \cup \{ t \}[/itex] where [itex]t \in X[/itex] is a single distinguished point.

i'm not really sure how to start.
i can visualise for example the n=1 case where [itex]D^1 \backslash S^0=[x-1,x) \cup (x,x+1][/itex] and [itex]S^1[/itex] is the unit circle so if we take t=x then [itex]D^1/S^0=[x-1,x+1][/itex]and i was thinking a homeomorphism here would be some sort of function along the lines of [itex]f(y)= \left( \cos{(y+1) \pi},\sin{(y+1) \pi} \right)[/itex]. i think this works as a homeomorphism but do i have to run through the steps of showing it's bijective and continuous and that the inverse is continuous?
 
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Answers and Replies

  • #2


Follow the hint. Think about for example, the The circle minis one point being homeomorphic to the line . You should be able to be explicit about this homeomorphism. It makes sense right? take away one point, and then "fold"" the surface out. Show that R is homeomorphic to any open interval (this is also easy and explicit).

Think about this for a while.
 
  • #3
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is the homeomorphism i gave above wrong and if so why? also, why am i considering the circle minus one point? surely i want the line minus one point unioned with one point to be homeomorphic to the enitre circle?
 

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