Re: A set containing a nonempty open interval

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.

>
> For example,
> A = [0,1] U [10,11]
> gives
> (A+A)/2 = [0,1] U [5,6] U [10,11]
> which contains no open intervals.

This contains the open interval (0,1) for example

>
> But if you stipulate x<>y you get
> (A+A)/2 = (0,1) U [5,6] U (10,11)
> which does contain open intervals.
>
> U is the union operator.
>
> --Keith Lewis klewis {at} mitre.org
> The above may not (yet) represent the opinions of my employer.
.

Relevant Pages

  • Re: A set containing a nonempty open interval
    ... >I'd like some hints on how to prove that, if a set S has positive ... Unless you specify x y, you might get all closed intervals. ... which does contain open intervals. ... Prev by Date: ...
    (sci.math)
  • Re: set theory question
    ... since my construction doesn't necessarily exhaust A. There could be ... Here's a set of replacement hints ... ... Argue that the intervals Uare pairwise disjoint. ... A set of pairwise disjoint open intervals must be countable. ...
    (sci.math)
  • Re: set theory question
    ... Hints are allowed to be vague. ... Elliot's) suggested strategy (I didn't read it the first time), ... Argue that the intervals Uare pairwise disjoint. ... A set of pairwise disjoint open intervals must be countable. ...
    (sci.math)
  • Re: A set containing a nonempty open interval
    ... The first set contains infinitely many open intervals. ... Amanda ... Prev by Date: ...
    (sci.math)
  • proof that cantor set is nowhere dense
    ... I want to see why the cantor set does not contain any open intervals. ... However, the length of the B_n's converge to zero, therefore C cannot ... Prev by Date: ...
    (sci.math)