Manifolds Challenge  

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Prove that the general n-sphere Sn = {x ∈ ℝn + 1 | x12 + ... + xn+12 = 1} is a n-dimensional manifold (Hint : Use the sets H<i = {x ∈ Sn | xi < 0}) and H>i = {x ∈ Sn | xi > 0}). Then, prove that any ellipsoid En = {x ∈ ℝn + 1 | ∑0 < i < n + 2 xi2 / ai2} is homeomorphic to Sn, and then give the charts on this ellipsoid that makes it a n-dimensional manifold (Hint : Given the homeomorphism f between the two, and the chart <U, x> of Sn, take x ∘ f).

 

Bonus : Prove that the product of two Hausdorff spaces is Hausdorff, and that the product of two spaces with a countable basis has a countable basis. Also prove that the product of two homeomorphisms is a homeomorphism (x : X → X', y : Y → Y' homeomorphisms, then x × y : X×Y → X'×Y' is also a homeomorphism). This will complete the proof that the product of two manifolds is a manifold.

That one guy that's always on

 
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My answer is shown below. Please do NOT view it unless you have given it a try!

That one guy that's always on

 
  
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