Tell me more ×
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.
up vote 10 down vote favorite
1

Can one endow the unit interval $[0,1]$ with a group operation to make it a topological group under its natural Euclidean topology?

share|improve this question
But one can endow $(0,1)$ with such a topology... – N. S. Dec 18 '12 at 20:07
@N.S.: With a topology or an operation? – Asaf Karagila Dec 18 '12 at 20:09
@AsafKaragila Ty, operation... I need my coffee :) – N. S. Dec 18 '12 at 20:12
Couldn't you define an operation on $Y=(-1,1)$ by the homeomorphism $f:\mathbb R\to Y,\ x\mapsto x/(|x|+1)$? It is possible to translate the operation on $\mathbb R$ onto $Y$, right? – Stefan H. Dec 18 '12 at 20:19
Any continuous bijection between $\mathbb R$ and $(0,1)$ should do the trick.. But there is probably something subtle happening, I think: the topology would be the Euclidian topology, but the uniformity which defines this topology is not the same.... – N. S. Dec 18 '12 at 20:27
show 1 more comment

1 Answer

up vote 17 down vote accepted

No. A topological group is homogeneous, and $[0,1]$ is not, since it has the two endpoints. (An open neighborhood of one of the endpoints, like $[0,1/2)$, is not homeomorphic to any open neighborhood of an interior point via a homeomorphism mapping $0$ to the interior point.)

share|improve this answer

Your Answer

 
discard

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.