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.


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

  • $\begingroup$ 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

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