Re: A set containing a nonempty open interval

On Fri, 26 Aug 2005 20:37:42 +0000 (UTC), klewis@xxxxxxxxxxxxxxxx
(Keith A. Lewis) wrote:

>A N Niel <anniel@xxxxxxxxxxxxxxxxxxxxx> writes in article <260820051428574700%anniel@xxxxxxxxxxxxxxxxxxxxx> dated Fri, 26 Aug 2005 14:28:57 -0400:
>>In article <denmfc$o5k$1@xxxxxxxxxxxxxxxxxxx>, Keith A. Lewis
>><klewis@xxxxxxxxxxxxxxxx> wrote:
>>
>>> "Amanda" <sca18@xxxxxxxxxxx> writes in article
>>> <1125067955.450481.88600@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> dated 26 Aug 2005
>>> 07:52:35 -0700:
>>> >
>>> >I'd like some hints on how to prove that, if a set S has positive
>>> >Lebesgue measure, then (A + A)/2 = {(x + y)/2 | x and y are in A}
>>> >contains a non-empty open interval.
>>>
>>> Unless you specify x <> y, you might get all closed intervals.
>>
>>By "contains a non-empty open interval" he means it has a subset which
>>is a non-empty open interval.
>
>Since every interval has a subset which is an open interval, I wonder why
>the word "open" is even in the question. Probably not important, I guess.

The closed interval [0,0] does not contain a (nonempty) open interval.

>--Keith Lewis klewis {at} mitre.org
>The above may not (yet) represent the opinions of my employer.


************************

David C. Ullrich
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