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quasi
Posts:
10,577
Registered:
7/15/05


Re: Uncountable?
Posted:
Apr 25, 2014 4:58 AM


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 jth 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 measurebased proof is both simple and natural.
In my opinion, it provides the underlying reason for the existence of such sets.
quasi



