CLASSIFICATION AND ORBIT EQUIVALENCE RELATIONS (Mathematical Surveys and Monographs 75)

By M. Džamonja. 2001

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A topological version of the Bergman property (2005, background): “Definition 1.3 A topological group G is said to have property (OB) if whenever G acts by isometries on a metric space (X, d), such that for every x ∈ X the function g ∈ G 7→ gx ∈ X is continuous, then every orbit is bounded.”
Complete analytic equivalence relations (2005, methodology): “In [11], Hjorth proves the existence of a complete (in fact “universal”) Π11 equivalence relation, using the norm and boundedness properties of this class.”
On the classification of Polish metric spaces up to isometry (2003, methodology): “Hjorth [1999] has shown that this equivalence relation is strictly above graph isomorphism.”