What's an example of an ordered set other than R that obeys the first 3 Suslin conditions?

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it's been more than a decade since i last thought about such things but would IR^n with the dictionary order work
- Yeah that should work I think. Maybe more interesting would be whether there exists an example which is not locally homeomorphic to IR (I think you're example still fulfills that). But I believe that is solved by using something like the long line and looking at e.g. ω₀\times 0. Is there an example that is nowhere locally homeomorphic to IR?
- IR as in irrationals? Would that be complete? Actually, I'm realising property 3 is kind of confusing as written in the wiki.
  
  IR as in the real numbers in fake blackboard bold :)
any continuous function that maps R onto something else, maybe?
- The range of such a function isn't guaranteed to to not be a real line again, though, and mapping it into a single point is an definite counterexample.
  
  Or do you mean some kind of ordering of the space of all such functions, maybe?

11 comments