Ball transitive ordered metric spaces

0. Introduction. If (X,≤) is a partially ordered set and A ⊆ X, then the decreasing hull d(A) of A in X is defined to be d(A) = {x ∈ X : x ≤ a for some a ∈ A}. If the poset X is not understood from the context, we may write dX(A). A subset A ⊆ X is a decreasing set if A = d(A). The intersection or union of any collection of decreasing sets in X is again a decreasing set in X. The increasing hull i(A) of a set A, and increasing sets are defined dually. The compliment of a decreasing set is an increasing set, and dually. A subset A ⊆ X is (order) convex if A = i(A)∩d(A), or equivalently, if a, b ∈ A and a ≤ c ≤ b, then c ∈ A. If (X,≤X) and (Y,≤Y ) are partially ordered sets, then the product order on X×Y is defined by (a, b) ≤ (c, d) if and only if a ≤X c and b ≤Y d. Unless otherwise noted, we will assume that the real line R carries its usual order and R carries the product order. If τ is a topology on X and ≤ is a partial order on X, then (X, τ,≤) is an ordered topological space. For A ⊆ X, the closed decreasing hull D(A) of A is the smallest closed decreasing set that contains A. The decreasing hull operator d : P(X) −→ P(X) and the closed decreasing hull operator D : P(X) −→ P(X) are both Kuratowski closure operators. Since cl(A) ⊆ D(A) and d(A) ⊆ D(A), we have d(cl(A)) ⊆ D(A) and cl(d(A)) ⊆ D(A), but in general, equality does not hold. It is known [6], however, that if A is compact, then d(A) = D(A), so that