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Homework Help: Function continuity in metric spaces

  1. Apr 10, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex](X,d_X)[/tex] and [tex](Y,d_Y)[/tex] be metric spaces and let [tex]f: X \to Y[/tex].

    2. Relevant equations

    Prove that the following statements are equivalent:

    1. [tex]f[/tex] is continuous on [tex]X[/tex],
    2. [tex]\overline{f^{-1}(B)} \subseteq f^{-1}(\overline{B})[/tex] for all subsets [tex]B \subseteq Y[/tex]

    3. The attempt at a solution

    I an prove that (1) leads to (2) but don't know how to show (2) leads to (1). Can you give me some hint?
  2. jcsd
  3. Apr 11, 2010 #2
    A function f: X -> Y between topological spaces is continuous if and only if the inverse image of every closed set in Y is a closed set in X.

    Let F be an arbitrary closed set in Y. Then F = cl(F), where cl(F) is the closure of F. Now apply (2) with F in place of B.
  4. Apr 21, 2010 #3
    Thanks, I didn't realize I should use this theorem and was trying to prove it using any point x and its neighborhood approach.
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