My Math Forum Intersection of an infinite number of open sets (induction)

 Real Analysis Real Analysis Math Forum

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 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 it´s 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?

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 $n\in\mathbb{N}$ 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, "$\infty\notin\mathbb{N}.$" Compare this to the statement "For any $n\in\mathbb{N}$ 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 $\{1\}+\{2\}+\{3\}+\cdots=\mathbb{N}.$
 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!

 Tags induction, infinite, intersection, number, open, sets

,

,

,

,

,

,

,

,

,

,

,

,

,

,

# infinite number of sets whose intersection is not open

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post fienefie Real Analysis 6 February 24th, 2015 02:14 PM veronicak5678 Real Analysis 1 May 1st, 2012 12:22 PM Mariogs379 Real Analysis 1 April 24th, 2012 09:02 PM veronicak5678 Real Analysis 4 April 19th, 2012 04:37 PM mizunoami Real Analysis 3 November 30th, 2011 06:06 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top