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July 26th, 2009, 06:48 AM  #1  
Newbie Joined: Jul 2009 Posts: 18 Thanks: 0  Intersection of an infinite number of open sets (induction)
Exercise 114 on Spivak's "Calculus on Manifolds" reads: Quote:
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? Thanks!  
July 26th, 2009, 08:04 AM  #2 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  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 
July 26th, 2009, 12:56 PM  #3 
Newbie Joined: Jul 2009 Posts: 18 Thanks: 0  Re: Intersection of an infinite number of open sets (induction)
Thanks!


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induction, infinite, intersection, number, open, sets 
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