Prove that the general n-sphere S^{n} = {x ∈ ℝ^{n + 1} | x_{1}^{2} + ... + x_{n+1}^{2} = 1} is a n-dimensional manifold (Hint : Use the sets H_{<}^{i} = {x ∈ S^{n} | x_{i} < 0}) and H_{>}^{i} = {x ∈ S^{n} | x_{i} > 0}). Then, prove that any ellipsoid E^{n} = {x ∈ ℝ^{n + 1} | ∑_{0 < i < n + 2} x_{i}^{2} / a_{i}^{2}} is homeomorphic to S^{n}, 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 S^{n}, 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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That one guy that's always on