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 SL be the set of isomorphism types of minimal L-structures. The elements of SL can be identified with ultrafilters of the Boolean algebra of quantifier-free L-sentences, and therefore one can define a Stone topology on SL. This topology on SL generalizes the topology of the space of n-marked groups. We introduce a natural ultrametric on SL, and show that the Stone topology on SL coincides with the topology of the ultrametric space SL iff the ultrametric space SL 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 SL, 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 spaces is precompact. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.