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Math Help - pathwise connected subset

  1. #1
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    pathwise connected subset

    Hi,

    Could someone help me prove the following:

    Let f:X \to Y be a continuos map between metric-spaces (X,d_{X}) and (Y,d_{Y}). Prove that if T \subseteq X is a pathwise connected subset in X, then the image f(T) \subseteq Y under f is a pathwise connected subset in Y.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    f:T\rightarrow f(T) is surjective so, for a,b\in f(T) there exist a',b'\in T such that f(a')=a and f(b')=b . As T is pathwise connected, there exists a path \gamma from a' to b' contained in T. Then, f\circ \gamma is a path from a to b contained in f(T) .


    Fernando Revilla
    Last edited by FernandoRevilla; January 12th 2011 at 04:08 AM.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by FernandoRevilla View Post
    f:T\rightarrow f(T) is surjective
    By the way surjective, I suppose you understand why f:T\rightarrow f(T) is surjective .
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