I'm stuck in this paper (Continuity Properties of Paretian Utility, Debreu 1964) for a month.
In the point two the author define a f-set as a nondegenerate subset A of S such that (a $\in$ A, b $\in$ A, a < b) $\Longrightarrow$ ({c $\in$ S | a $\leqslant$ c $\leqslant$ b} is a finite subset of A)
in my head the image of this definition it's something like this:
after that, this sentence came out and i really don't know what's going on
"Consequently, an f-set necessarily has one of the four forms;
- finite
- a_0 < a_1 < a_2 < ...
- ... < a_-2 < a_-1 < a_0
- ... < a_-1 < a_o < a_1 < ..."
The cases 2, 3 and 4 aren't infinite subsets? how theses are possibles forms of a f-set?
The condition for an $f$-set can be restated more plainly as "$A$ is an $f$-set if for each pair of elements of $A$ there are finitely many other elements of $A$ between them." It doesn't require $A$ to be finite.
In each of the cases listed at the end of your post, if you pick any two elements of those sets, there are only finitely many elements between them.