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Infinite subset is dense

  1. Sep 3, 2005 #1
    My problem is:

    Consider an infinite set [itex]X[/itex] with the finite complement topology. I want to show that any infinite subset [itex]A[/itex] of [itex]X[/itex] is dense in [itex]X[/itex].

    Now, I can show that every point of [itex]X[/itex] is a limit point of [itex]A[/itex].

    Can this help me in any way to show that [itex]A[/itex] is dense in [itex]X[/itex]. Or could someone provide me with an alternative method.

    By the way, my understanding of dense is that a subset of a set is dense if its closure coincides with the set.
     
  2. jcsd
  3. Sep 3, 2005 #2

    Hurkyl

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    Let me rephrase your question:

    If every point of X is a limit point of A, then is A dense in X?
     
  4. Sep 3, 2005 #3
    Yes, that is my question.
     
  5. Sep 3, 2005 #4

    Hurkyl

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    So what do the definitions say?
     
  6. Sep 3, 2005 #5
    Wait a sec, if every point of X is a limit point of A, then A is dense in X!
     
  7. Sep 3, 2005 #6

    matt grime

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    SO, A is any infinite subset and you want to find the smallest closed set containing A. Since there are three kinds of closed set:

    the empty set

    a set containing a finite number of points

    all of X


    isn't the answer obvious? I mean, whichof those can contain A, given A is infinite?
     
  8. Sep 3, 2005 #7
    is it all of X?
     
  9. Sep 3, 2005 #8

    matt grime

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    well, can you name a finite (or empty set) that contains an infinite subset?
     
  10. Sep 3, 2005 #9
    nope, I can't.

    So am I correct in thinking that by proving that every point of [itex]X[/itex] is the limit point of [itex]A[/itex], then [itex]A[/itex] is dense?
     
  11. Sep 3, 2005 #10

    Hurkyl

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    That is correct -- the point of my first post was to eliminate the unnecessary stuff, hoping you could see this when that's all that's left. :smile:

    However, the approach matt has mentioned is a much easier way to do this problem... and is a fairly important theme to understand in general.
     
  12. Sep 3, 2005 #11
    I agree, Matt's method was MUCH easier.
     
  13. Sep 3, 2005 #12

    HallsofIvy

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    Once again- look at the DEFINITION of "dense"!
     
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