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Math Help - open? closed? both?

  1. #1
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    open? closed? both?

    Consider f(x) = 1/(1+x^2) as a function from R to R with the usual metric. Are the following statements true or false (give proofs or counterexamples):

    1. f(A) is open for all open sets A in R
    2. f(A) is open for all closed sets A in R

    thanks!
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  2. #2
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    1. Let A=R. Then A is open and f(A) = (0,1].

    2 is obviously false (take A to be a singleton). Did you mean f(A) is closed for closed sets A? If so, use the same example as (1).
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  3. #3
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    Quote Originally Posted by maddas View Post
    1. Let A=R. Then A is open and f(A) = (0,1].

    2 is obviously false (take A to be a singleton). Did you mean f(A) is closed for closed sets A? If so, use the same example as (1).
    2. thanks!
    nope... f(A) open if A is closed

    1. I'm not sure that proves anything
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  4. #4
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    Quote Originally Posted by maddas View Post
    1. Let A=R. Then A is open and f(A) = (0,1].

    2 is obviously false (take A to be a singleton). Did you mean f(A) is closed for closed sets A? If so, use the same example as (1).
    2. thanks!
    nope... f(A) open if A is closed

    1. I'm not sure that proves anything
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  5. #5
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    Is (0,1] open?
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  6. #6
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    Quote Originally Posted by maddas View Post
    Is (0,1] open?
    no, but it is not closed either
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  7. #7
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    Quote Originally Posted by swallenberg View Post
    no, but it is not closed either
    oh, I just got it... sorry I was focusing on the second one! Thanks!!!
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