Onn-ordered sets and order completeness
Lino Gutierrez Novoa · Pacific Journal of Mathematics · 1965
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 w-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 ά-orders in the "flats" of an ^-ordered set and the fact that (as it happens for chains) the topological structure is Hausdorff.The idea of Dedekind cut is extended to ^-ordered sets and the notions of strong-completeness, completeness, and conditional completeness are introduced.It is shown that the S n sphere is s-complete when considered as an ^-ordered set.It is also proved that E n , the ^-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.IL n-Ordered sets* An element of the cartesian product X n+1 of a set X will be called an ^-simplex and denoted by σ n = (s 09 s 19 , s n ) where s t 19, t k [ is the h + k -1 -simplex ( s 09 s 19 , s h9 1 0 , t 19 , t k \ and will be denoted by | σ h , τ k |.An n-ordered set is a pair (X, φ n ), where X is a set and φ n is a function from | X n \ to the set {-1, 0, 1} and which satisfies A 1 and A.A 10 -For every \ σ n \ e \ X n [ <p n \σ n \ =φ n \ σ n \.Before stating A 2 we introduce the following notation:, s n \ φ n \ t 0 9 1 1 9 , ί^, s 0 , t 1+19 •••,<"!