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Math Help - Show that T is surjective

  1. #1
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    Show that T is surjective

    Let (X,d) be a compact metric space, and let T: X \longrightarrow X be a continuous map satisfying the expansion property:
    d(T(x),T(y)) \geq d(x,y) for all x,y \in X.
    Prove that T is surjective.

    Hey I've difficulty starting this question. How do I go about doing it?

    Thanks in advance.
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  2. #2
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    Re: Show that T is surjective

    my thought is this:

    choose an open cover of ε-balls for X. since X is compact, it has a finite subcover. consider images under T of this subcover. show they form a cover of X.

    (hint: for every U in our sub-cover, we have U contained in T(U) because....?)
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  3. #3
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    Re: Show that T is surjective

    Another way would be to prove the result by contradiction. Suppose that the point u\in X is not in the range of T. The range of T is closed (by compactness) so there exists \delta>0 such that d(u,Tx)\geqslant\delta for all x in X.

    Now consider the sequence u,Tu,T^2u,T^3u,\ldots. Use the non-contracting property of T to show that any two points in this sequence are at a distance at least \delta apart. Therefore the sequence cannot have a convergent subsequence, which contradicts the compactness of X.
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  4. #4
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    Re: Show that T is surjective

    U is not necessarily contained in T(U). See for example X is the standard sphere, T(p)=-p is the antipodal map.

    Quote Originally Posted by Deveno View Post
    my thought is this:

    choose an open cover of ε-balls for X. since X is compact, it has a finite subcover. consider images under T of this subcover. show they form a cover of X.

    (hint: for every U in our sub-cover, we have U contained in T(U) because....?)
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  5. #5
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    Re: Show that T is surjective

    Quote Originally Posted by xxp9 View Post
    U is not necessarily contained in T(U). See for example X is the standard sphere, T(p)=-p is the antipodal map.
    that's right...for x in U, T(x) might not even be in U. my bad.
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  6. #6
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    Re: Show that T is surjective

    sorry deleted
    Last edited by xxp9; November 19th 2011 at 06:21 PM.
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