Categories of compact zero-dimensional spaces
by
R. W. Knight
University of Oxford

Corresponding to any set of axioms in a countable language of first order logic is a category of separable metric spaces, the spaces of types, which are compact by the Tarski-Maltsev compactness theorem. Ehrenfeucht proved that, if the theory is complete, then it has exactly one countable model if and only if all these spaces are finite. They have been further studied by Vaught and others.

It seems reasonable to assert that the number of countable models of a set of axioms is determined by this category of topological spaces. We discuss this question, and the topological theory emerging from it.

Date received: June 15, 2004