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The space of minimal structures
Author(s) -
Belegradek Oleg
Publication year - 2014
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201300012
Subject(s) - mathematics , ultrametric space , compact space , hausdorff space , topology (electrical circuits) , compact open topology , metric space , general topology , bounded function , isomorphism (crystallography) , topological space , space (punctuation) , urysohn and completely hausdorff spaces , discrete mathematics , class (philosophy) , combinatorics , pure mathematics , hausdorff dimension , interpolation space , hausdorff measure , functional analysis , computer science , mathematical analysis , chemistry , crystal structure , operating system , biochemistry , gene , crystallography , artificial intelligence
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 coincides 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 < ω ). 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 spaces is precompact.