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Math Help - Show f is a bijection

  1. #1
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    Show f is a bijection

    For any p ∈B
    f: (A x {p})→ A
    Show f is a bijection (one-to-one and onto)
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  2. #2
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    Re: Show f is a bijection

    Quote Originally Posted by Kiefer View Post
    For any p ∈B
    f: (A x {p})→ A
    Show f is a bijection (one-to-one and onto)
    You have not defined the function.
    You have told us the domain and range.
    BUT what is f(A\times\{p\})=~?
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  3. #3
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    Re: Show f is a bijection

    This is part of a topology proof: For any p2 ∈ X2
    π1|(x1x{p2}): (X1 x {p2})→ X1 is a homeomorphism
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  4. #4
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    Re: Show f is a bijection

    Quote Originally Posted by Kiefer View Post
    This is part of a topology proof: For any p2 ∈ X2
    π1|(x1x{p2}): (X1 x {p2})→ X1 is a homeomorphism
    You cannot just throw us down into the middle of a proof.
    If you mean a mapping f:A\times\{p\}\to A defined by (a,p)\mapsto a, which is the natural projection of A\times\{p\} onto A then there is nothing to prove. It is a bijection.
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  5. #5
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    Re: Show f is a bijection

    This is part of a topology proof: For any p2 ∈ X2
    π1|(x1x{p2}): (X1 x {p2})→ X1 is a homeomorphism

    Sorry, I phrased that poorly. Above is the whole topology proof as given to me in class. In my original post, I was just trying to simplify notation. Part of being a homeomorphism is being a bijection, and that is the part I need help proving. Here is an outline for the proof:

    Proof: Consider any p2 ∈ X2
    Claim 1: g: (X1 x {p2})→ X1 is a bijection
    Proof: Let g=π1|(x1x{p2}) (just to simplify notation)
    Claim 1.1: If g(p)=g(q), then p=q (ie: g is injective or one-to-one)
    Proof: Let g(p)=g(q)
    [Can I simply say g(g(p))=g(g(q)) and therefore p=q?]
    Claim 1.2: For every x∈ X1, there exists p∈ (X1 x {p2}) such that g(p)=x (ie: g is surjective or onto)
    Proof: consider any x∈ X1
    [Again, this seems like a simple proof, but I could use some help here]
    Therefore, g: (X1 x {p2})→ X1 is injective from Claim 1.1
    And g: (X1 x {p2})→ X1 is surjective from Claim 1.2
    Therefore, g: (X1 x {p2})→ X1 is a bijection
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  6. #6
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    Re: Show f is a bijection

    no, you have to PROVE that if g(x1,p) = g(x2,p) that (x1,p) = (x2,p).

    however, note that g(x,p) = x, for all x in X.

    so if g(x1,p) = g(x2,p), it is immediate that x1 = x2.

    since x1 = x2, and p = p, (x1,p) = (x2,p) by the definition of equality in the cartesian product.

    you also have to prove that for ANY x in X, there is some y in X x {p} with g(y) = x.

    i suggest y = (x,p).

    **********
    the fact that g:X x {p} → X given by g(x,p) = x is a bijection is, admittedly, rather trivial. p is just "along for the ride", all we do for any x in X is just add (_,p) to it.

    the proof that g is a HOMEOMORPHISM is more involved. here, the explict TOPOLOGIES on Xx{p} and X need to be SPECIFIED.

    presumably, you are using the subspace (relative) topology of Xx{p} relative to the PRODUCT topology on XxY (where p is a point of Y), where some explicit topologies are given on X and Y.

    although you do not say so, it may be the case that X = Y = R, the real numbers with the standard topology (defined by the base of open intervals under the euclidean metric).

    in any case, it IS clear that g is a continuous bijection. what is less trivial is that g-1 is also continuous.
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