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

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!

• Crossposted to math.se: math.stackexchange.com/questions/3534793/…
– bof
Feb 5 '20 at 6:57
• See my answer on math stackexchange here. Feb 5 '20 at 22:16
• E.g. the so-called double arrow is a classic separable non-metrisable compact LOTS. Feb 5 '20 at 22:46
• @WholeFood, I rolled your edit back to the previous version because the post was too cryptic in the modified version, whereas the original one is more easily readable. For instance, I had no idea what the acronym "LOTS" was. Jul 13 at 17:26

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.
• The general condition for LOTS's is having a $G_\delta$ diagonal or a $\sigma$-locally countable base. Theory from the 1970's when LOTS's where studied more. Feb 5 '20 at 22:18