Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is there an associative operation $\star \colon \mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0} $ wich also happens to be a metric on $\mathbb{R}_{\geq 0}$? Furthermore is there a monoid/group structure with such an operation? Would be the group topological in respect to the metric-induced topology?

(The usual metric given by $d(x,y) = \lvert y - x \rvert$ for $x, y \in \mathbb{R}_{\geq 0}$ is not associative since $2 = d(d(1,2),3) \neq d(1,d(2,3)) = 0$.)

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Yes, it is true.

Please check this link http://mathoverflow.net/questions/16214/is-there-an-associative-metric-on-the-non-negative-reals .


share|improve this answer
Oh nice. I couldn't find anything on this site, but I didn't look on mathoverflow. –  k.stm Oct 8 '12 at 9:25

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.