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Topic: Uncountable?
Replies: 9   Last Post: Apr 26, 2014 2:13 PM

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quasi

Posts: 10,257
Registered: 7/15/05
Re: Uncountable?
Posted: Apr 25, 2014 4:58 AM
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William Elliot wrote:
>quasi wrote:
>>William Elliot wrote:
>>>
>>>Let C be a collection of open subsets of R that cover Q.
>>>Can S = R - \/C ever be uncountable?


Yes, as I outlined in my earlier reply (in your first thread).

>Let qj be the j-th rational and Uj = (qj - 2^-j, qj + 2^-j).
>C = { Uj | j in N } covers Q. The measure of \/C <= 2 and
>the measure of R is oo. Thus most of R is outside of \/C.


The above is a standard argument -- I'm surprised you weren't
aware of it. It's usually discussed in Elementary Analysis,
then again in Real Analysis, and still once more in a first
course in Topology.

>Can it be shown directly without measure theory that R - \/C
>is uncountable?


Perhaps, but why bother?

The measure-based proof is both simple and natural.

In my opinion, it provides the underlying reason for the
existence of such sets.

quasi



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