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Math Help - Images of Compact and Connected Sets

  1. #1
    mathisnotforme101
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    Post Images of Compact and Connected Sets

    Does anybody know an example of a continous map F: R --> R and a closed Subset B C R such that f(b) is not closed.
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  2. #2
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    No, because by definition every continuous image of an open set is open, whence every continuous image of a closed set is closed.
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  3. #3
    Newbie Jay Gatsby's Avatar
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    Quote Originally Posted by mathisnotforme101 View Post
    Does anybody know an example of a continous map F: R --> R and a closed Subset B C R such that f(b) is not closed.
    Let f(x)=\frac{1}{1+x^2} and B=[0,\infty). It is clear that B is a closed set, whereas f(B)=(0,1] is not closed.
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  4. #4
    Newbie Jay Gatsby's Avatar
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    Quote Originally Posted by julien View Post
    No, because by definition every continuous image of an open set is open, whence every continuous image of a closed set is closed.
    Julien, you have the definition backward. The inverse image of an open set under a continuous function is open. Likewise, the inverse image of a closed set under a continuous function is also closed.
    Last edited by Jay Gatsby; December 5th 2006 at 02:33 PM.
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