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Math Help - closure

  1. #1
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    closure

    how
    Last edited by mathmad; January 3rd 2010 at 06:41 AM.
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  2. #2
    Senior Member Shanks's Avatar
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    The conclusion is not true if f is not continous function.
    If the contious condition is assumed, then it follows immediately from the continuousity of continous function.
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  3. #3
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    I am sorry

    The function f is continuous F:X tends to Y
    and X and Y are two complete metric spaces

    Thank you
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  4. #4
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    Suppose that V is a subset. By the continuity of f we have f^{-1}(\overline{f(V)}) is closed in X (f^-1 from Y to X maps closed/open sets to closed/open sets resp.).
    Notice that f(V) \subset \overline{f(V)} \implies V\subset f^{-1}(\overline{f(V)}). As the closure is the smallest closed superset of V we have \overline{V}\subset f^{-1}(\overline{f(V)}). Now take f of both sides to get the result.

    You don't need completeness (or the fact that it is a metric space).
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  5. #5
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    ........
    Last edited by mathmad; November 24th 2009 at 11:23 AM. Reason: ........
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