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:


1. $X$ is a compact linearly ordered topological space (compact LOTS)
2. $X$ 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!


Thank you! 



Compact, densely ordered spaces
One of my favorite spaces has this property: the *extended long ray*.


First, the long ray itself: this is just the space $L$ gotten by pasting together $\omega\_1$-many copies of $[0, 1)$ in the natural way. Formally, $L$ is the lexicographic order on $\omega\_1\times[0, 1)$, with both viewed as linear spaces in the natural way.


Now, $L$ is densely ordered, but obviously $L$ is not compact (although any *countable* open cover admits a finite subcover, and every sequence contains a convergent subsequence; and, as an unrelated nice property, every proper initial segment of $L$ is homeomorphic to $[0, 1)$). However, if we add a point at the end, we get the extended long ray $L^\*$.


$L^\*$ is still densely ordered, and unlike $L$ is compact; in fact, if I remember correctly, it is both the Stone-Cech compactification *and* the one-point compactification of $L$. $L^\*$ is a nice counterexample to a bunch of things, as is $L$ itself; in particular, the differentiable structures admitted by $L$ are many and nasty.




---


I suspect the wonderful book "Counterexamples in Topology" (Steen & Seebach) has many more examples of such spaces.


