Re: -- rational distances

On 2 Oct, 06:20, quasi <qu...@xxxxxxxx> wrote:
On Thu, 02 Oct 2008 01:16:12 -0400, quasi <qu...@xxxxxxxx> wrote:
On Wed, 1 Oct 2008 21:40:42 -0700 (PDT), smn <smnewber...@xxxxxxxxxxx>
wrote:
On Oct 1, 9:14 pm, quasi <qu...@xxxxxxxx> wrote:
Does there exist a closed, nonempty subset S of R^2 such that d(p,S)
is in Q for all points p in Q^2?

quasi

S=R^2 smn

Yep, I forgot to bar that.

Ok, consided it barred ...

Does there exist a closed, nonempty proper subset S of R^2 such that
d(p,S) is in Q for all points p in Q^2?

No, that can also be easily achieved.

For example, let S = {(x,y) in R^2 | y <=0}.

This seems to work because you don't have to take square roots, and
anyway require that the frontier of S is only made of rational points,
which I suppose means the frontier stays always parallel to one or the
other axis.

Ok, here's a fixed version ...

Does there exist a nonempty compact subset S of R^2 such that d(p,S)
is in Q for all points p in Q^2?

As soon as you make S bounded, I wouldn't see how you can fulfil the
required property at all, because you'll have anyway to take square
roots for the distance from almost all points...

-LV
.

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