Re: -- rational distances

Timothy Murphy <tim@xxxxxxxxxxxx> writes:

quasi wrote:

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?

It wasn't clear to me if a proof of the result has
been given.
But it seems to me much easier than the arguments
given.

The distance d(p,S) from a point p to S is a
continuous function of p.
It is not constant, so it cannot always be
rational.

Maybe I mis-read the question?

Yes, you did. Note the "p in Q^2". There are plenty
of continuous
functions from Q^2 to Q.

but there are even more reals than rationals !!


--
Robert Israel
israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics
http://www.math.ubc.ca/~israel
University of British Columbia Vancouver,
BC, Canada

regards

tommy1729
.

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