markw...@yahoo.com wrote:
> On Jun 11, 2:57 pm, Hatto von Aquitanien <a...@AugiaDives.hre> wrote:
>> This, however, tells me that I could define a metric as a rational-valued
>> function:

>> http://planetmath.org/encyclopedia/AlternativeDefinitionOfMetricSpace...

>> Is there any fundamental problem in doing so in order to enable the
>> completion of Q to R?

> That article you said you were going to read sometime later?

> The
> answer was sitting right there. A metric space is any space such that
> every two points A, B has a distance AB, such that
> (1) AA = 0
> (2) AB > 0 if A, B are distinct points
> (3) AB = BA
> (4) AB + BC >= AC.

> Now, for rationals, A, B take the distance AB to be the absolute value
> of their difference |A-B|.

> Is (1) true? |A-A| = 0. Yes.
> Is (2) true? |...| is always non-negative. If |A-B| = 0, then A = B.
> So, by exclusion |A-B| > 0 if A and B are different numbers. So, yes.
> Is (3) true? |A-B| = }-(A-B)| = |B-A|. So, yes here too.
> Finally, is (4) true? |A-B| + |B-C| = |A-C| if (A-B) and (B-C) have
> the same sign (or if either A = B or B = C); otherwise |A-B| + |B-C| =
> |A+C-2B| > |A-C| if (A-B) and (B-C) has opposite signs. Hence,
> property (4) is true, too. Either AB + BC = AC (if B lies on or
> between A and C), or else AB + BC < AC.

> To see, by the way, that the latter equality |A+C-2B| > |A-C| *does*
> actually follow if (A-B) and (B-C) have opposite signs, consider the
> ratio r = (C-B)/(A-B). Then |A+C-2B| = |A-B+C-B| = (1+r)|A-B|, while |
> A-C| = |A-B+B-C| = |(1-r)(A-B)| = |1-r||A-B|. Since 1+r > |1-r| for
> all positive r, then the result follows.

> Thus, the rationals with this distance function satisfies the defining
> properties of a metric space.