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

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

    HallsofIvy

    User Avatar
    Science Advisor

    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

    Bacle2

    User Avatar
    Science Advisor

    Or, you can apply DeMorgan to an intersection of opens that is not open, like (0,1/n).
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook