0
$\begingroup$

I'm trying to prove that lower limit and upper limit topologies on R are homeomorphic.

However, it is clear that the identity is not an homeomorphism because, for example, $[0,1)$ is open in the lower limit topology, but not in the upper limit topology, and this is true for any half-open set; it is open in one, but not in the other, so the homeomorphism has to map such half open intervals to the half intervals (in the opposite sense), but I cannot think any such map; even after thinking on it for 2 weeks.

Of course, it has to also map open intervals to open intervals too.

$\endgroup$
3
$\begingroup$

$f(x) = -x$ is a homeomorphism, If $a < b$, $f^{-1}[(a,b]] = [-b, -a)$ etc.

$\endgroup$
  • $\begingroup$ Well, thanks a lot. Apparently, I forgot one of the two trivial maps; identity and the maps that send every element to its "inverse". $\endgroup$ – onurcanbektas Nov 17 '18 at 12:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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