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.
