Our aim here is to study this property. The exclusion of the vector 0 is not essential in what follows. It turns out that many of the other properties of root systems of Lie algebras depend only on property P and are thus shared by a much wider class of vector systems. In the statements to follow it is assumed that all vectors considered come from a real Euclidean space Vof finite dimension n. 1.1 Let S be finite and have property P, and let A be a subset with property P. For each x in S, assume that at most one (resp. exactly one, at least one) of x and -x is in A. Then there is an ordering of the space V such that A is contained in (resp. A is, A contains) the set of positive elements of S. Harish-Chandra [6, Lemma 4] proves the second part of this result for Lie algebra root systems, however, using nontrivial properties of Lie algebras. Borel and Hirzebruch [2, pp. 471-473] give a geometric proof of the second and third parts, but then revert to Lie algebra techniques to prove the first part, all for Lie algebra root systems. All of these authors make a somewhat stronger assumption on A than property P, namely, if x, y e A and x + y e S, then x + y e A. Our proof of 1.1 depends on a preliminary result which may have some independent interest. 1.2. Let B be finite and have property P, and assume that x e B implies -x q B. Then ther-e is an ordering of the space V such that all elements of B are positive. The real numbers of the form k 112 (k, I positive integers) show that the assumption of finiteness in 1.1 (or 1.2) can not be dropped. It can, however, be weakened thus. 1.1'. In 1.1 replace the assumption of finiteness by the assumption that 0 is not a point of accumulation of S. In 1.2 we can go further, in terms of a weakening of property P.