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Infinite union of closed sets that isn't closed?

  1. Nov 10, 2011 #1
    So I have to find an infinite union of closed sets that isn't closed. I've thought of something that might work:

    [itex]\bigcup[0,x][/itex] where [itex]0\leq x<1[/itex]. Then, [itex]\bigcup[0,x] = [0,1)[/itex], right?
  2. jcsd
  3. Nov 10, 2011 #2


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    Yes, that is correct. 1 is not in that union because it is not in [0, x] for any x<1. If y< 1, however, there does exist x> y and y is in such [0, x]. Therefore the union contains all of [0, 1).
  4. Nov 11, 2011 #3


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    Or, you can apply DeMorgan to an intersection of opens that is not open, like (0,1/n).
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