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July 26th, 2009, 06:48 AM   #1
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Intersection of an infinite number of open sets (induction)

Exercise 1-14 on Spivak's "Calculus on Manifolds" reads:

Quote:
Prove that the intersection of two (and hence of finitely many) open sets is open. Give a counterexample for infinitely many open sets.
I proved that the intersection of two open sets is open and then used induction to generalize it to any number of open sets (it is true for two sets. then, assuming itīs true for n sets, the intersection of n+1 open sets is just the intersection of n open sets (wich is an open set) and another open set, so it is an open set). The problem is I don't think this rules out an infinite number of sets.

It's easy to find a counterexample:

Let In = (0, 1 + 1/n). Then the intersection of all In (with n between 1 and infinity) is I = (0,1], which is not open.

So, what's wrong with my use of induction, here?

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July 26th, 2009, 08:04 AM   #2
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Re: Intersection of an infinite number of open sets (induction)

The induction argument proves that for any open sets, the intersection of these sets is also open. It does not say anything about an infinite collection of sets, since, to put it crudely, ""

Compare this to the statement "For any finite sets, the union of these sets is also finite". This statement can be proved with induction in a similar way to your problem, but it would be nonsensical to suggest that the union of an infinite collection of finite sets must also be finite (consider
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July 26th, 2009, 12:56 PM   #3
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Re: Intersection of an infinite number of open sets (induction)

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