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latentcorpse
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If D^n is the unit n ball in Euclidean n-space. i.e.
D^n = \{ x \in \mathbb{R}^n : ||x|| \leq 1 \}
and S^n is an n-sphere.
how do i show that D^n / S^{n-1} is homeomorphic to S^n?
there's a hint suggesting i first of all try the n=1,2,3 cases. where X/Y= X \backslash Y \cup \{ t \} where t \in X is a single distinguished point.
i'm not really sure how to start.
i can visualise for example the n=1 case where D^1 \backslash S^0=[x-1,x) \cup (x,x+1] and S^1 is the unit circle so if we take t=x then D^1/S^0=[x-1,x+1]and i was thinking a homeomorphism here would be some sort of function along the lines of f(y)= \left( \cos{(y+1) \pi},\sin{(y+1) \pi} \right). 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?
D^n = \{ x \in \mathbb{R}^n : ||x|| \leq 1 \}
and S^n is an n-sphere.
how do i show that D^n / S^{n-1} is homeomorphic to S^n?
there's a hint suggesting i first of all try the n=1,2,3 cases. where X/Y= X \backslash Y \cup \{ t \} where t \in X is a single distinguished point.
i'm not really sure how to start.
i can visualise for example the n=1 case where D^1 \backslash S^0=[x-1,x) \cup (x,x+1] and S^1 is the unit circle so if we take t=x then D^1/S^0=[x-1,x+1]and i was thinking a homeomorphism here would be some sort of function along the lines of f(y)= \left( \cos{(y+1) \pi},\sin{(y+1) \pi} \right). 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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