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Math Help - real projective plane

  1. #1
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    real projective plane

    I'm working on a proof to show there exists an embedding of the real projective plane P R^2 in R^4.
    The initial setup is as follows:
    Let S^2 denote the unit sphere in R^3 given by S^2 = {(x, y, z) ∈ R3 : x^2 + y^ 2 + z ^2 = 1}, and let
    f : S^2 → R^4 be defined by f (x, y, z) = (x^2 − y ^2 , xy, yz, zx).
    I'm trying to show that f determines a continuous map F: P R^2 → R^4 where P R^2 is the real projective plane,
    then show that F is a homeomorphism onto a topological subspace of R^4 .
    I think it's easy to see that f(x1,y1,z1)=f(x2,y2,z2) implies (x1,y1,z1)=+/-(x2,y2,z2). But I don't know how to figure out the whole proof completely. Could anyone please give me a hint? Any input is appreciated!
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  2. #2
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    Quote Originally Posted by nngktr View Post
    I'm working on a proof to show there exists an embedding of the real projective plane P R^2 in R^4.
    The initial setup is as follows:
    Let S^2 denote the unit sphere in R^3 given by S^2 = {(x, y, z) ∈ R3 : x^2 + y^ 2 + z ^2 = 1}, and let
    f : S^2 → R^4 be defined by f (x, y, z) = (x^2 − y ^2 , xy, yz, zx).
    I'm trying to show that f determines a continuous map F: P R^2 → R^4 where P R^2 is the real projective plane,
    then show that F is a homeomorphism onto a topological subspace of R^4 .
    I think it's easy to see that f(x1,y1,z1)=f(x2,y2,z2) implies (x1,y1,z1)=+/-(x2,y2,z2). But I don't know how to figure out the whole proof completely. Could anyone please give me a hint? Any input is appreciated!
    So you have shown that f is a well-defined injective function from PR^2 to R^4. Continuity is easy, because each coordinate of f(x,y,z) is a continuous function of (x,y,z). Finally, a continuous bijection from a compact space to a Hausdorff space is automatically bicontinuous. So f is a homeomorphism.
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