Infinite union of closed sets that isn't closed?

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  • #1
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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?
 

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  • #2
HallsofIvy
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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).
 
  • #3
Bacle2
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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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