Compact, densely ordered spaces

Question

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!

Answer 1

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.

Answer 2

The linear orders which are compact in the order topology are precisely the complete totally ordered sets $X$. By complete I mean that every subset of $X$ has a least upper bound including $\emptyset$ and $X$. If $R\subseteq X$ has no least upper bound, then let $S$ be the set of all upper bounds of $R$. Then $\bigcup_{r\in R}\{x\in X|x<r\}\cup\bigcup_{s\in S}\{x\in X|x>s\}$ is an open cover with no finite subcover.

For the converse, assume $X$ is a complete totally ordered set. Let $\mathcal{U}$ be a non-principal ultrafilter on $X$. Then for each $a\in X$ either $\{x\in X|a\leq x\}\in\mathcal{U}$ or $\{x\in X|a\geq x\}\in\mathcal{U}$. Let $A=\{x\in X|a\geq x\}\in\mathcal{U}$ and let $B=\{x\in X|a\leq x\}\in\mathcal{U}$. Then $A$ is downwards closed and $B$ is upwards closed and $\{A,B\}$ is a partition of $X$. Therefore there is some $x\in X$ where if $a\leq x$, then $a\in A$ and if $a\geq x$, then $a\in B$. Let $(a,b)$ be an interval around $x$. Then $a\in A,b\in B$, so $(a,1],[0,b)\in\mathcal{U}$, so $(a,b)=(a,1]\cap[0,b)\in\mathcal{U}$. Therefore $X$ is compact since every non-principal ultrafilter converges.

Answer 3

A compact Suslin line would be another important example although they only exist in some models of set theory. Dedekind completions of suitable Aronszajn lines give you more ZFC examples.