Math Is Fun Forum

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Baire's category theorem

I found this to be a rather interesting proof.  Use Baire's category theorem to prove the following:

If

is a collection of open dense sets in a complete metric space

, then

is dense.

Give a counter example to show that "open" is a required property, without using the axiom of choice.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Baire's category theorem

Bump!  I'll be posting the really cool solution to this problem by tonight unless someone tells me not to.  So if you want to work on it, let me know.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."