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Math Help - Continuity open sets

  1. #1
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    Continuity open sets

    How are these equivalent to the regular epsilon-delta definition of continuity?

    (i) Let  f: M \to N . The pre-image of each closed set in  N is closed in  M .

    (ii) The pre-image of each open set in  N is open in  M .
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  2. #2
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    Quote Originally Posted by shilz222 View Post
    How are these equivalent to the regular epsilon-delta definition of continuity?
    (i) Let  f: M \to N . The pre-image of each closed set in  N is closed in  M .
    (ii) The pre-image of each open set in  N is open in  M .
    In a general space they are not equivalent. But they are in a metric space and in particular \Re^1.
    Note that \left| {x - x_0 } \right| < \delta \quad  \Leftrightarrow \quad x \in \left( {x_0  - \delta ,x_0  + \delta } \right).
    In \Re^1 the basic open set is an open inteveral.
    \left( {x_0  - \delta ,x_0  + \delta } \right) is the a basic open inteveral, in fact it is a ball centered at x_0 with radius \delta.

    Can you work out the details?
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  3. #3
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    That book shows that the 2 are equivalent by using sequences. They use the definition of sequence:  d(x,p) < \delta \implies f( f(x), f(p)) < \epsilon .

    I haven't covered balls yet, but I think now that I follow that the book is doing. It seems more intuitive with sequences.
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  4. #4
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    Oh I got it. You have to use the rule that the complement of an open set is closed right?
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  5. #5
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    Quote Originally Posted by shilz222 View Post
    Oh I got it. You have to use the rule that the complement of an open set is closed right?
    Yes, that will work.
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