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Math Help - Quick Topology question

  1. #1
    MHF Contributor Mathstud28's Avatar
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    Quick Topology question

    I was reviewing some old problems and I came across a problem which I found interesting. The problem although not very difficult was neat because it was something that was implicitly implied often but never proven, the question was

    Prove that any open interval in \mathbb{R} is homeomorphic to any other interval in \mathbb{R}
    My solution was simple, I conjectured that the homeomorphism was linear and found that the function f(x)=\frac{d-c}{b-a}x+\frac{bc-ad}{b-a} satisfies the correct conditions. The book has just answers and validated the above function, but I see a problem that I wish someone else could help me with. Don't we need connectivity for this homeomorphism to apply? Otherwise how can we assume that all intermediate values are taken?

    Thanks for any help
    Alex
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Sorry, problem solved. A diminuitive footnote in the back stated that the sets may be assumed to be connected intervals.
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  3. #3
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    Opalg's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    A diminutive footnote in the back stated that the sets may be assumed to be connected intervals.
    But does the book assume that intervals have to be finite? If not, then you also need to consider intervals of the form (a,∞), (∞,a) and indeed (∞,∞).
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Opalg View Post
    But does the book assume that intervals have to be finite? If not, then you also need to consider intervals of the form (a,∞), (∞,a) and indeed (∞,∞).
    It does assume they are finite, it is in the notation bit in the beginning. Thank you though, the book has already given examples of these homeomorphisms. Thanks though Opalg
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