Infinite union of closed sets that isn't closed?

[autre]

So I have to find an infinite union of closed sets that isn't closed. I've thought of something that might work:

$\bigcup[0,x]$ where $0\leq x<1$. Then, $\bigcup[0,x] = [0,1)$, right?

 

[HallsofIvy]

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).

 

[Bacle2]

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