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Thread: Open set

  1. #1
    Feb 2009

    Open set

    A map f:\mathbb{R} \to \mathbb{R} is called open if f(A) is open for every open
    subset A of R. Show that every continuous open map of R into
    itself is monotonic.
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  2. #2
    MHF Contributor

    Aug 2006
    Because this function is continuous then on any interval [a,b] it has a high point and a low point: \left( {\exists \left\{ {c,d} \right\} \subseteq \left[ {a,b} \right]} \right)\left( {\forall x \in \left[ {a,b} \right]} \right)\left[ {f(d) \leqslant f(x) \leqslant f(c)} \right].
    Suppose that \left\{ {c,d} \right\} \subseteq \left( {a,b} \right) then f\left[ {\left( {a,b} \right)} \right] = \left[ {f(d),f(c)} \right] which is contradictory to f being open.
    A similar argument leads to the conclusion that the maximum and minimum must happen at the endpoints of any closed interval.

    Suppose that there are points a < b < c\;. \mathrel\backepsilon  .\;f(a) < f(b)\;\& \;f(c) < f(b).
    But that would mean that the maximum on the interval [a,c] is not at endpoint.
    The same contradiction would follow if a < b < c\;. \mathrel\backepsilon  .\;f(a) > f(b)\;\& \;f(c) > f(b).
    In other words f must be monotonic.
    Last edited by Plato; Mar 6th 2009 at 03:21 PM.
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