Infinite intersection of open sets

[michonamona]

I understand that the finite intersection of open set is open, but is it true that the infinite intersection of open set is closed? or is it possible for it to be open as well?

Thank you,

M

 

[lanedance]

would depend on the sets...

say its the intersecyion of (-1-n, 1+n), n>=1, then it will be open = to (-1,1)

but if its (-1/n, 1/n), then it will be closed = [0]

so i think you can still find infinite intersections of open sets that are open, but you can't guarantee and infinite itersection of open sets is open like you can in the finite case

 

[michonamona]

Did you mean (-1+(1/n), 1-(1/n))? where n is a positive integer. As n approaches infinity, the boundaries of the set gets closer and closer to -1 and 1 but never actually touches them, thus the intersection is open.

Thanks for the examples.

 

[lanedance]

no i meant (-1/n, 1/n)

but pick any point e close to 0, you can always choose n=N, such that e is outside the set for (-1/N, 1/N), so in the limiting process the interesection becomes only the single point zero

its the infinite limit that makes these tricky