In this paper, the notion of
an n-ordered set is introduced as a natural generalization of that of a totally ordered
set (chain). Two axioms suffice to describe an n-order on a set, which induces
three associated structures called respectively: the incidence, the convexity,
and the topological structures generated by the order. Some properties of
these structures are proved as they are needed for the final theorems. In
particular, the existence of natural k-orders in the “flats” of an n-ordered
set and the fact that (as it happens for chains) the topological structure is
Hausdorff.
The idea of Dedekind cut is extended to n-ordered sets and the notions of
strong-completeness, completeness, and conditional completeness are introduced. It is
shown that the Sn sphere is s-complete when considered as an n-ordered set. It is
also proved that En, the n-dimensional euclidean space, fails to be s-complete or
complete, but that it is conditionally complete. It is also proved that every
s-complete set is compact in its order topology but that the converse is not true.
These results generalize classical ones about the structure of chains and
lattices.
