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Math Help - continuous on metric space

  1. #1
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    continuous on metric space

    Let f: (M,d)\to (N,p). Prove that f is continuous if and only if f(\text{cl}(A))\subset \text{cl}(f(A)) for every A \subset M if and only if f^{-1}(\text{int}(B))\subset \text{int}(f^{-1}(B)) for every B \subset N.
    Give an example of a continuous f such that f(\text{cl}(A)) \ne \text{cl}(f(A)) for some A \subset M.

    I have shown that the closure case is true, i need some help on the interior case and the example.
    I was told to use the fact that inverse image of every open set is open
    Last edited by Opalg; November 15th 2011 at 01:07 AM. Reason: tidied up LaTeX
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  2. #2
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    Re: continuous on metric space

    Quote Originally Posted by wopashui View Post
    Let f: (M,d)\to (N,p). Prove that f is continuous if and only if f^{-1}(\text{int}(B))\subset \text{int}(f^{-1}(B)) for every B \subset N.
    Note that if O is an open set in N then \text{int}(O)=O.
    If f(x_0)\in O then x_0\in (f^{-1}(O))=f^{-1}(\text{int}(O))\subseteq \text{int}(f^{-1}(O))
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