# A question about dimension and connectedness in order topologies

Does there exist a totally disconnected topological space T whose topology is the order topology of a linearly ordered set and whose (small inductive) dimension is equal to 1? There exist topological spaces which satisfy all the other conditions, but I do not know of any whose topology is an order topology.

• That $\operatorname{ind}(X) \le 1$ (trivial) and even $\dim(X) \le 1$ (slightly less so) for ordered spaces was well-known, I'm sure, so that sort of trivialises the dimension theory before you even start thinking about it. – Henno Brandsma Oct 13 '14 at 3:51