It is well known that $[a,b]_\mathbb{R}$ is connected, while $[a,b]_\mathbb{Q}$ is not.
I'm wondering, what conditions on a totally ordered set $T$ imply that for all $a,b \in T$, it holds that $[a,b]_T$ is connected? Weak conditions are especially interesting to me; an "iff" statement would be just fabulous.
Unless I have made a mistake, the following hold in any totally ordered set.
- Every connected subset is an interval.
- If furthermore every set of the form $[a,b]_T$ is connected, then every interval is connected.
Hint: Prove that a total ordered set $(X,<)$ is connected iff $(X,<)$ is dense and Dedekind complete. That $(X,<)$ must be dense and Dedekind is clearly necessary. For the other direction suppose $A\subsetneq X$ is open and nonempty. Pick $a\in A$ such that $a$ is not the supremum of $X$. There are two cases. If $\{x\in X:a<x,x\notin A\}\neq\emptyset$, put $b=\sup\{x\in X:a<x,x\in A\}$, and show $b\in A^c$ and that $b$ is not in the interior of $A^c$. Otherwise, $\{x\in X:x<a,x\notin A\}\neq\emptyset$, then take $b=\inf\{x:x<a,x\in A\}$ instead.