Hyperreal numbers $^*\mathbb{R}$ form a totally ordered field that is a proper extension of the real numbers (i.e. non-isomorphic to real numbers $\mathbb{R}$). It is obvious that $^*\mathbb{R}$ has the following properties, related to the Suslin's problem:
- $^*\mathbb{R}$ does not have a least nor a greatest element;
- the order on $^*\mathbb{R}$ is dense (between any two distinct elements there is another);
but I am not sure if the other two conditions are satisfied in $^*\mathbb{R}$:
- Is the order on $^*\mathbb{R}$ complete? (in the sense that every non-empty bounded subset has a supremum and an infimum)
- Is every collection of mutually disjoint non-empty open intervals in $^*\mathbb{R}$ countable? (this would be the countable chain condition for the order topology of $^*\mathbb{R}$),
More simply, I am asking: Are hyperreal numbers an example of a Suslin line? Why yes or why no?
No non-Archimedean (let alone hyperreal) field is complete: if $F$ is non-Archimedean, the set $I_F$ of infinitesimal elements of $F$ is bounded but has no supremum in $F$.
Two quick endnotes:
First, note my language above. Despite common misuse, there is no such thing as "the" hyperreals. Rather, there is a notion of a hyperreal field, of which there are many examples. Hyperreal fields are non-Archimedean fields satisfying an additional strong property called "transfer" (and for that matter, the exact version of transfer demanded can vary from text to text, so care is required here).
Second, I've left open the question of ccc-ness. This is because it's not needed for your main question in light of my point above re: completeness, and it's more complicated. In fact, any hyperreal field constructed via ultrapowers (which is the most common approach) will fail to be ccc, but this takes a bit of effort to show.