Re: Uncountable many reals without Cantor

From: Dave Seaman (dseaman_at_no.such.host)
Date: 11/30/04 

Date: Tue, 30 Nov 2004 21:05:47 +0000 (UTC)

On Tue, 30 Nov 2004 22:36:38 +0200, Toni Lassila wrote:
> On Tue, 30 Nov 2004 19:51:26 +0000 (UTC), Dave Seaman
><dseaman@no.such.host> wrote:
>>On 30 Nov 2004 10:27:24 -0800, Daryl McCullough wrote:

>>> 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.
>>
>>That property is called "sequential compactness", and you're right that
>>it doesn't particularly help. The key here is the Heine-Borel theorem,
>>which says that closed, bounded intervals in R^n are compact.

> I don't know whether you were just being unclear but "every Cauchy
> sequence converges" is certainly not the definition of a compact set,
> it is the definition of a complete metric space.

> The definition of sequential compactness is that every sequence in the
> set has a subsequence that converges in the set.

Yes, sorry. I was being sloppy. In a metric space, compactness is
equivalent to sequential compactness, and both hold iff the space is
complete and totally bounded. But compactness is the property that's
needed in the measure theory proof, one way or another. 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

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