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ion of coming before in time or in place or in magnitude—which we denote by the curved inequality sign —? is said to be a linear order, if the following uniqueness and transitivity properties hold : Uniqueness: For every pair of elements a, b of 0 one and only one of the three relationships a—ib, a = b, b—ia holds. Transitivity: If a—lb and H e , then a—lc. Examples of linear order are the set of integers arranged in the order of ascending magnitude; the set of real numbers arranged in order of descending, magnitude; the set of positive integers where the rank relationship is defined as follows: the integer wHw 2 , if the number of prime factors of n% is less than tha t of w2; and in case the number of prime factors is the same, then n\—<n<L, if ni<fi2 in the ordinary sense. The first two orders are not normal, tha t is to say, not every subset has a first element, not even the set itself, but

Methods in point sets and the theory of real functions