What's an example of an ordered set other than R that obeys the first 3 Suslin conditions?
Suslin's problem - Wikipedia
https://en.wikipedia.org/wiki/Suslin%27s_problem#Formulation
https://en.wikipedia.org/wiki/Suslin%27s_problem#Formulation
Ah. IR^n is separable, though. By Cantor's mentioned theorem (which is irritatingly not cited) it must be order-isomorphic to IR if it meets the 3 conditions and is separable.
There has to be a simple example, though, right? Suslin added the fourth condition. I thought of the long line, but that seemed tricky for a couple of reasons.
I didn't mean IR^n with its usual topology. I meant IR^n with the order topology for the dictionary order. IIANM you can construct an uncountable set of pairwise disjoint open intervals in this topology so it can't have a countable dense subset. But as I said it's been years since I touched a topology book.
IIANM you can construct an uncountable set of pairwise disjoint open intervals in this topology
Hmm. Do you have a construction in mind?
I think you could just take an open interval in the order topology and then create a collection by turning the first dimension into a parameter. IIANM for each value of the parameter you'd get an open set, they'd be pairwise disjoint, and there'd be uncountably many of them.