The axiomatizability of topological prevarieties

We investigate first-order axiomatic descriptions of naturally occurring classes of Boolean topological structures (these structures can have operations and relations, and carry a compatible compact Hausdorff topology with a basis of clopen sets). Our methods utilize inverse limits and ultraproducts of finite structures. We illustrate the range of possible axiomatizations of these classes with applications of our methods to Boolean topological lattices, graphs, ordered structures, unary algebras and semigroups. For example, whereas the class of all k-colorable graphs is known to be axiomatizable by universal Horn sentences, we find the class of continuously k-colorable Boolean topological graphs is not even first-order axiomatizable.