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My analysis book includes a theorem that states:

1. The union of any number of open sets is an open set.

2. The intersection of a finite number of open sets is an open set.

I follow the proof of property one, however in part of the proof of property two my book directs me to a picture of nested sets and says: "A glance at the above figure will convince you that the intersection is an open interval". The book then goes on to say: "Note that this is where the argument breaks down for infinite intersections: if there were an infinite number of intervals the intersection might not be an open interval. QED".

I can perhaps buy into the proof that the intersection of a finite number of open sets is an open set, but this in no way convinces me of anything pertaining to the intersection of an infinite number of open sets!

Perhaps I'm just dense, but I really feel the need for explicit proof that the intersection of an infinite number of open sets may not be open.

My other thought on this was that maybe there was no way to prove or disprove the infinite case due to the ambiguity of it; that is because the intersection 'might' or 'might not' be open.

This general concern also reminds me of the notion of raising negative one to an infinite power. There is no way to say if the result is a negative or positive number.

If anyone could try to help me understand my problem of infinite intersections I would be very appreciative! Also, if anyone has some insight on how to interpret and deal with mathematical ambiguities in general, I would love to hear what you have to say on the topic!

thanks