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Math Help - Prove that the n-dimensional real projective space RP^n is compact

  1. #1
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    Prove that the n-dimensional real projective space RP^n is compact

    Prove that the n-dimensional real projective space \mathbb{RP}^n is compact.

    I wouldn't know where to start.

    Further, I have another question.

    I need to prove that R with its cofinite topology is compact and sequentially compact. Compact seems easy, I did it like this:

    Let (U_n) be an open cover for R, no select one (non-empty) of these open sets, say U_k0. Now, R\U_k0 is obviously finite so we can write it like {a_1,...,a_n}. Now, every one of these elements is in an open set. So, U_k0 together with these sets is an open cover.

    Is this correct?
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  2. #2
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    The real projective space is compact becuase it is the quotient of a compact space (the n dimensional unit sphere).
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