Is a totally ordered, separable and connected topological space metrizable (in the order topology)?


If we relax the assumption of connectedness, I know the counterexamples, but if we have a linear continuum that is also separable, can we say it is metrizable?


Thanks!



Is a totally ordered, separable and connected topological space metrizable (in the order topology)?
Yes, because it is regular and has a countable base, namely the family of open intervals with the ends at the dense countable set. The connectedness guarantees that each open interval $(a,b)$ is not empty and hence contains a point from the countable dense set.


