The space of minimal structures

Abstract

For a signature L with at least one constant symbol, an L-structure is called minimal if it has no proper substructures. Let S_L be the set of isomorphism types of minimal L-structures. The elements of S_L can be identified with ultrafilters of the Boolean algebra of quantifier-free L-sentences, and therefore one can define a Stone topology on S_L. This topology on S_L generalizes the topology of the space of n-marked groups. We introduce a natural ultrametric on S_L, and show that the Stone topology on S_L coincide with the topology of the ultrametric space S_L iff the ultrametric space S_L is compact iff L is locally finite (that is, L contains finitely many n-ary symbols for any n). As one of the applications of compactness of the Stone topology on S_L, we prove compactness of certain classes of metric spaces in the Gromov-Hausdorff topology. This slightly refines the known result based on Gromov's ideas that any uniformly totally bounded class of compact metric spases is precompact.