# Prove that lower limit and upper limit topologies on R are homeomorphic

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.