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Math Help - Homeomorphisms

  1. #1
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    Homeomorphisms

    Let f: (M_1,d_1)\to(M_2,d_2) and G(f)=\{(x,f(x))|x\in M_1\}\subseteq M_1\times M_2 and let \tilde f: (M_1,d_1)\to(G(f),d_p). Consider \tilde f(x)=(x,f(x)) and d_p((x_1,x_2),(y_1,y_2))=\sqrt{d_1(x_1,y_1)^2+d_2(  x_2,y_2)^2} for all (x_1,y_1),(x_2,y_2)\in M_1\times M_2. Prove that \tilde f is a homeomorphism.

    In order to be a homeomorphism we need bijectivity for \tilde f and continuity for \tilde f and \tilde f^{-1}, is there a faster way to solve this?
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  2. #2
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    Re: Homeomorphisms

    \text{Let } \pi : (G(f), d_p) \rightarrow (M_1, d_1) \text{ by } \pi(x,y) = x \ \forall (x,y) \in G(f).

    \text{To show it's a bijection, show } (\tilde{f} \circ \pi)(x,y) = (x,y) \ \forall \ (x,y) \in G(f),

    \text{and } (\pi \circ \tilde{f})(x) = x \ \forall \ x \in M_1.

    \text{Note that } (G(f), d_p) \text{ is the subspace with the induced metric from the Euclidean-type product metric } d_p^*

    \text{on }(M_1, d_1) \times (M_2, d_2) \cong (M_1 \times M_2, d_p^*). \text{(Perhaps you'd want to prove the identity is a homeo?)}

    \text{Thus } \pi = proj_1|G(f) \text{ of } proj_1 :  (M_1 \times M_2, d_p^*) \rightarrow (M_1, d_1), \text{ and so } \pi \text{ is continuous.}

    \text{Note that } \tilde{f} \text{ is the subspace restriction of a composition of the diagonal map (always continous) with a continuous map: }

    \Delta \ : \ (M_1, d_1) \rightarrow (M_1, d_1) \times (M_1, d_1), \ x \mapsto (x, x)

    id \times f \ : \ (M_1, d_1) \times (M_1, d_1) \rightarrow (M_1, d_1) \times (M_2, d_2) \cong (M_1 \times M_2, d_p^*),

    (id \times f)(x,y) = (x, f(y)).

    \text{This means that } \tilde{f} = (id \times f) \circ \Delta \text{ is continuous.}

    \text{If you're unconvinced, direct proofs for any part of the justification of continuity of } \pi \text{ and } \tilde{f} \text{ will be straightforward.}
    Last edited by johnsomeone; October 18th 2012 at 10:57 PM.
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