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Homework Help: Analysis: the closure of a set is closed?

  1. Sep 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove or disprove the following statement:

    The closure of a set S is closed.



    2. Relevant equations

    Definition of closure: set T is the closure of set S means that T is the union of S and the set of limit points of S.

    Definition of a closed set: set S is closed means that if p is a limit point of S then p is in S.


    3. The attempt at a solution

    So, the closure of set S-- call it set T-- contains all the elements of S and also all the limit points of S. Then, T must have limit points that is contained in it.

    I can't prove that all the limit points of T is in T. I can only prove that all the limit points of S is in T. Help?
     
  2. jcsd
  3. Sep 2, 2010 #2

    radou

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    Homework Helper

    You can make this very simple.

    The definition of a closure of a set A: the closure of A is the intersection of all closed sets containing A.

    In a topological space, what is the intersection of any number of closed sets?
     
  4. Sep 2, 2010 #3

    Dick

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    That the limit points of S are in T is directly given by the definition of closure. That's not much of a start to the proof. If you want to do this without radou's alternative definition of 'closure', pick p to be a limit point of T. Can't you show that's also a limit point of S? Stating the definition of 'limit point' might help.
     
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