During my work with order preserving homeomorphisms, I got interested in the double arrow space and, subsequently, in the lexicographic square. I would really like to find examples of spaces like these two. Specifically, I am looking for a space $X$ with the following properties:
- $X$ is a compact linearly ordered topological space (compact LOTS)
- $X$ is densely ordered, that is for all $x,y\in X$ if $x <y$ then there exists $z\in X$ such that $x<z<y$. (I think one can say that in this case $X$ "has no jumps".)
Besides the spaces mentioned above, closed real intervals also satisfy these conditions. I think that maybe the pseudo-arc is another example, although I am still trying to understand its construction.
Any other examples or references would be highly appreciated!