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Topic: Subset of the square
Replies: 9   Last Post: Nov 25, 2000 12:55 PM

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Hugo van der Sanden

Posts: 234
Registered: 12/8/04
Re: Subset of the square
Posted: Nov 23, 2000 8:37 PM
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Colin Hayman wrote:
> > Joseph levine wrote:
> > > Does there exist a subset of the unit square such that all horizontal
> > > lines intersect the set in countably many points but all vertical
> > > lines intersect the set in uncountably many points?

> >
> > ((x, y): x rational, 0 < x,y < 1) would seem to be a trivial solution.

> That was one of my first thoughts, but consider the vertical line x =
> 1/pi. Any point on that line has an irrational x-coordinate between 0 and
> 1 (namely 1/pi), so it doesn't contain any points from your subset -- and
> zero is about as countable as it gets.

Thanks, I suspected I had to be missing something.


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