A sheaf of real numbers

Abstract

θ is a σ-topology on an arbitrary non-empty set I. Each open set U ε θ is a "truth-value" and we write [ α ] for the truth-value of the formula α. R* is the set of locally constant partial functions on I with codomain R (the set of real numbers), and it becomes the complete θ-set (or sheaf) R* when we define [ f = g ] = { i | i Є I; f, g defined at i; f(i) = g(i)} for all f, g Є R*. R* may be thought of as a "generalised" set of real numbers and our aim here is to characterise the sheaf R* in the way that the phrase "complete ordered field" characterises the set R.