(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

a) If U and V are open sets, then the intersection of U and V (written U \cap V) is an open set.

b) Is this true for an infinite collection of open sets?

2. Relevant equations

Just knowledge about open sets.

3. The attempt at a solution

a) Let U and V be open sets, x be in U \cap V, and a,b be real numbers. Then there exists open intervals (a1,b1) in U and (a2,b2) in V such that x is in (a1,b1) and x is in (a2,b2). Now let (a,b) = (a1,b1) \cap (a2,b2). Since x is in (a1, b1) and x is in (a2,b2), then x is in (a,b) and by definition is an open interval. Moreover, since (a,b) is a subset of U \cap V, then U \cap V is an open set.

Is there anything I'm missing from this proof, like proving that the intersection (a1,b1) \cap (a2,b2) is an open interval? Or is that overdoing it..?

b) I know this isn't true (the intersection from n=1 to infinity of all open sets (-1/n, 1/n) = {0}, which is not an open set), but again I feel like I'm missing something - specifically proving that {0} is the intersection of all of those open sets, and proving {0} is not open.

I know these questions are incredibly straightforward, but my professor is kind of a stickler on the little details, especially the ones that seem pretty obvious. Thanks for the help!

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# Homework Help: Intersection of open sets

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