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Prove that for any collection {Oα} of open subsets of ℝ, [itex]\bigcap[/itex] 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 [itex]\ni[/itex] A. Let O= [itex]\bigcap[/itex] 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

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 [itex]\ni[/itex] A. Let O= [itex]\bigcap[/itex] 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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