of results relating properties of the order topology to the completeness of X: If the order topology on X is connected, X is complete. X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is two points a and b in X with a < b such that no c satisfies a < c < b.) X is complete if and only if every bounded set that is closed in the order topology is compact. A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervals of real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line). There are order-preserving homeomorphisms between these examples. Sums of orders For any two disjoint total orders and , there is a natural order on the set , which is called the sum of the two orders or sometimes just : For , holds if and only if one of the following holds: and and and Intuitively, this means that the elements of the second set are added on top of the elements of the first set. More generally, if is a totally ordered index set, and for each the structure is a linear order, where the sets are pairwise disjoint, then the natural total order on is defined by For , holds if: Either there is some with or there are some in with , Decidability The first-order theory of total orders is decidable, i.e. there is an algorithm for deciding which first-order statements hold for all total orders. Using interpretability in S2S, the monadic second-order theory of countable total orders is also decidable. Orders on the Cartesian product of totally ordered sets In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product of two totally ordered sets are: Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order. (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order. (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct