Infinite totally ordered set with all non-empty subsets containing both endpoints? I'm wondering if there are any infinite sets which can be totally ordered in such a way that all non-empty subsets contain a maximal and minimal element.
It is clear to me that this is impossible in discrete orderings and in the standard bounded dense orderings like $[0,1]$, but is it known that this is impossible in general?

Prove that a partial order with a top and bottom, for which
every not empty subset contains the top and the bottom, has
exactly one point.
