Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

R(=real Nos)

  1. Apr 21, 2009 #1
    Given that f is a function from R(=real Nos) to R continuous on R AND ,A any subset of R,IS THE closure of f(A) ,a closed set??
     
  2. jcsd
  3. Apr 22, 2009 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Re: closure

    The closure of any set is by definition a closed set. I think you should rephrase your question.
     
    Last edited: Apr 22, 2009
  4. Apr 22, 2009 #3
    Re: closure

    Yes, you right thank you. But if we define a set to be closed if its complement is open,
    how then we prove its closure to be a closed set??
     
    Last edited: Apr 22, 2009
  5. Apr 23, 2009 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Re: closure

    What definition of "closure of A" are you using?
     
  6. Apr 23, 2009 #5
    Re: closure

    what? by definition the closure of a set A is the smallest closed set that contains A.
     
  7. Apr 23, 2009 #6
    Re: closure

    I presume the OP had in the mind the definition that the closure of S is the union of S and the set of its limit points. In this case:

    Denote by S' the closure of S. Then we wish to show that S' is closed. Suppose x is in the complement of S'. Then x is not in S and is not a limit point of S. So there is an open ball around x that doesn't intersect S. This open ball cannot contain any limit point of S since if y is inside it, then there is a smaller ball centered at y contained in the bigger - and so there is an open ball around y that doesn't intersect S, so y is not a limit point of S. It follows that the open ball around x does not intersect S'. Therefore the complement of S' is open; so S' is closed.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: R(=real Nos)
  1. R integrable ? (Replies: 1)

  2. Real Analysis (Replies: 0)

  3. Statistics in R (Replies: 2)

  4. Gradient of r (Replies: 1)

Loading...