Discreteness and continuum assumptions

A two-event chain has no intermediate event. More generally, an order with finite intervals cannot interpolate unconditionally between all distinct comparable points: repeated interpolation would produce infinitely many elements inside one interval.

Even an interval basis and density do not imply a manifold. The ordered rational line has both, but every neighborhood is countable and therefore cannot be homeomorphic to a nonempty open subset of ${\mathbb{R}}^d$, $d\ge 1$. The distinction between a discrete order and its possible continuum approximation is also fundamental in causal-set theory Bombelli.