Diagonalization of continuous matrices as a representation of intuitionistic reals

We use topological models of intuitionistic analysis to answer some of the recent questions of Kadison [7] concerning diagonalization of matrices of continuous functions. Let X be a compact Hausdorff space, and let %(X) be the ring of continuous real functions on X. In the topological interpretation of intuitionistic analysis over X [3], %(X) represents the internal set of (Dedekind) reals. Thus we can address the questions from [7] simply by intuitionistic examination of elementary linear algebra. We show that for O-dimensional spaces X, diagonalization of y1 x rz symmetric matrices over q(X) is characterized by requiring that any two disjoint open F, sets have disjoint closures, i.e. X is an F-space [4,5]. In fact, we obtain this result as a consequence of a stronger, internal intuitionistic result. Internally, the condition states that the ordering 6 on Dedekind reals is a linear ordering. When we first obtained these results in March 1983, we were aware of the slightly prior related work of W. Zame, who used the methods of analysis. Our main point here is to establish these results as a simple application of logic.