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Math Help - Continous function

  1. #1
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    Continous function

    Let f: (X1,d1) -> (X2, d2) be a continous function. Let C, a subset of X2 be compact.

    Is f^-1(C) ("f inverse") compact in X2?

    Having trouble finding a counterexample...
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  2. #2
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    Quote Originally Posted by math_help View Post
    Let f: (X1,d1) -> (X2, d2) be a continous function. Let C, a subset of X2 be compact.

    Is f^-1(C) ("f inverse") compact in X2?

    Having trouble finding a counterexample...
    I'm assuming that you mean X_1

    Consider the function

    f: \mathbb{R}\to \mathbb{R} with the usual metric

    f(x)=1

    The one point set \{ 1 \} is compact in the range. (It is closed and bounded)

    but f^{-1}(1)=\mathbb{R} this set is unbouned and not compact.
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  3. #3
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    What do you mean by saying I assume you mean X1?
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  4. #4
    Behold, the power of SARDINES!
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    Quote Originally Posted by math_help View Post
    What do you mean by saying I assume you mean X1?
    Is f^-1(C) ("f inverse") compact in X2?

    f^{-1}(C) \subset X_1 not X_2
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