Is the Intersection of Open Sets Always Open?

[mathanon]

Prove that for any collection {Oα} of open subsets of ℝ, \bigcap Oα is open.


I did the following for the union, but I don't see where to go with the intersection of a set.

Here's what I have so far:

Suppose Oα is an open set for each x \ni A. Let O= \bigcap Oα. Consider an arbitrary x in O. By definition of O, x is in O, and O is open by hypothesis. So x is an interior point of Oα

 

[LCKurtz]

Prove that for any collection {Oα} of open subsets of ℝ, \bigcap Oα is open.

What if $O_n = (-\frac 1 n, \frac 1 n)$?