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

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Math Lover

Posts: 65
Registered: 4/9/14
Re: Uncountable?
Posted: Apr 25, 2014 4:35 AM
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In article <Pine.NEB.4.64.1404250003140.11979@panix2.panix.com>,
William Elliot <marsh@panix.com> wrote:

> Let C be a collection of open subsets of R that cover Q.
> Can S = R - \/C ever be uncountable?


Yes, since the cover can be as "small" as you want (by that I mean
its measure). Number the elements of Q somehow: q1, q2, q3 ... qn...
(this can be done since Q is countable). Cover q1 by an interval
of size 1, q2 by an interval of size 1/2, and so on; the total
measure of the cover is <= 2 (<= because some intervals may overlap),
so your "S" is of infinite measure, hence uncountable (although
it does not contain any open interval).

Cheers, L.



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