Re: Uncountable many reals without Cantor

From: Daryl McCullough (daryl_at_atc-nycorp.com)
Date: 11/30/04 

Date: 30 Nov 2004 10:27:24 -0800

Dave Seaman says...

>>>(*) sum length(I_n) >= 1.
>>>
>>>How do you _prove_ that?
>
>> Assume the opposite, put the intervals end to end etc. This kind of
>> thing is proven in the beginning of any Real Variables text, e.g.
>> Royden. Where do you see a problem?
>
>Sounds like you want to use induction on the number of intervals.
>Problem is, induction works only if the number of intervals is finite.
>
>Hint: that's where compactness comes in.

Just for clarification of this comment.

Compactness for a topological space means the following: X is compact
if for any collection of open sets U_i whose union is X, there is a
finite subcollection whose union is also X. So a closed interval of
finite length is compact, but an open interval is not.

I was confused at first because I was misremembering the definition
of compactness. I remembered a definition along the lines of "X is
compact if every Cauchy sequence converges". That doesn't directly
help much. 

--
Daryl McCullough
Ithaca, NY

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