Is the Intersection of Open Sets Always Open?

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SUMMARY

The intersection of any collection of open subsets of ℝ, denoted as \(\bigcap O_{\alpha}\), is not necessarily open. The discussion highlights that while individual sets \(O_{\alpha}\) are open, their intersection may include boundary points that are not interior points, thus failing to meet the definition of an open set. A specific example provided is \(O_n = (-\frac{1}{n}, \frac{1}{n})\), which illustrates that as \(n\) approaches infinity, the intersection converges to the closed set \{0\}, confirming that the intersection is not open.

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mathanon
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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
 
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mathanon said:
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)##?
 

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