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Thread: continuous on metric space

  1. #1
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    continuous on metric space

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

    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; Nov 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 $\displaystyle f: (M,d)\to (N,p)$. Prove that f is continuous if and only if $\displaystyle f^{-1}(\text{int}(B))\subset \text{int}(f^{-1}(B)) $ for every $\displaystyle B \subset N$.
    Note that if $\displaystyle O$ is an open set in $\displaystyle N$ then $\displaystyle \text{int}(O)=O$.
    If $\displaystyle f(x_0)\in O$ then $\displaystyle x_0\in (f^{-1}(O))=f^{-1}(\text{int}(O))\subseteq \text{int}(f^{-1}(O))$
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