The complexity space of a valued linearly ordered set

By a valued linearly ordered set (a VLOS for short), we mean a pair (X,φ) such that X is a linearly ordered set and φ is a strictly increasing (= positive monotone) nonnegative real valued function. Clearly, any VLOS is a valuation space. Each VLOS (X,φ) generates a linear weightable quasi-metric dφ on X whose conjugate is order preserving. We show that the Smyth completion of (X, dφ) also admits the structure of a VLOS. On the other hand, M. Schellekens introduced in 1995, the theory of complexity spaces to develop a topological foundation for the complexity analysis of programs. Here, we introduce the so-called complexity space of a VLOS (X,φ) and discuss some of its properties. In particular, we show that it is weightable and preserves Smyth completeness of (X, dφ).We apply this complexity approach to the measurement of real numbers and discuss some advantages of our methods with respect to those that use the classical Baire metric. Mathematics Subject Classification (2000): 54E35, 54F05, 68Q25.