# Is there a nontrivial LOTS that is connected and totally path disconnected?

By nontrivial LOTS I mean a linearly ordered space that contains more then one point. Being totally path disconnected means that every path in the space is constant.

A connected linearly ordered topological space (LOTS) may not be path-connected, the ordered square being an standard counterexample. But can it be totally path disconnected?

I don't have any idea. Any help appreciated.

• See here: mathoverflow.net/questions/213208/…. May 18 at 12:10
• @ElchananSolomon How is that a linearly ordered space? May 18 at 12:15

Let $$X = [0,1]^\omega$$ lexicographically ordered. It is a standard result that $$X$$ is connected.
Any non-trivial interval of $$X$$ contains an uncountable pairwise disjoint family $$(\;](x_1, ..., x_n, x, 0, 0, ...), (x_1, ..., x_n, x, 1, 1, ...)[\;)_{0 \le x \le 1}$$ of open, non-empty intervals, hence cannot be homeomorphic to a subset of the reals. Therefore any path in X is constant.