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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.

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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.

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