Like the title: Are there any examples of connected, dense total order spaces which are not separable?
I've still haven't made much progress with this. I know that they are order complete from being order dense and connected, but anything else escapes me.
Either an example or a refutation that such spaces exists would be most enlightening.
$[0,1] \times [0,1]$ in the lexicographic order (and its induced topology), (so $(x,y) < (u,v)$ iff ($x < u$ or $x=u$ and $y<v$) ) is an example of a connected (and thus order dense) compact ordered space that is not separable. Removing $(0,0)$ and $(1,1)$ creates a non-compact example. Another non-compact (but countably compact) space is the long line e.g.
A well-known theorem says that if $X$ is an ordered space that is connected (i.e. order dense and complete) and has no minimum nor maximum and is separable, then $X$ is homeomorphic (even order isomorphic) to the real line. This inspired the Suslin problem (whether we could weaken "separable" to ccc here, see wikipedia. And this gave rise to nice results in set theory and independence results. Both examples above are non-ccc such spaces. Under certain axioms (like $\diamondsuit$) we could use a Suslin line as a ccc example, which are even closer to the reals. In ZFC we cannot find such a space.