The space of minimal structures

TL;DR

This paper explores the topological and metric properties of the space of minimal structures in model theory, establishing conditions for compactness and applications to metric space classes.

Contribution

It introduces a natural ultrametric on the space of minimal structures and characterizes when the Stone topology coincides with this ultrametric topology, linking to local finiteness.

Findings

Stone topology on minimal structures is compact iff the language is locally finite.

The ultrametric topology coincides with the Stone topology under compactness conditions.

Application to the compactness of classes of metric spaces in Gromov-Hausdorff topology.

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.