On the classification of Polish metric spaces up to isometry

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Complete analytic equivalence relations (2005, background): “Another very similar Kσ equivalence relation was already known to be complete (see Kechris [13]).”
Completeness of the isomorphism problem for separable C*-algebras (2013, background): “Gao and Kechris deduce it from the fact that for every separable complete metric space X there is an isometric copy Z(X) of X in U such that F (U) ∋ X 7→ Z(X) ∈ F (U) is Borel and for every X,Y ∈ F (U) if X and Y are isometric, then there is an isometry ϕ ∈ Iso(U) such that ϕ′′Z(X) = Z(Y ).”
The complexity of classifying separable Banach spaces up to isomorphism (2006, background): “Also, by the result of Becker and Kechris quoted above, we only have to consider Polish spaces X and continuous actions α : G×X → X. View X as a closed subset of X0 and associate to each x ∈ X the element Fx ∈ Z defined by Fx = {(y, g) ∈ X0 ×G ∣∣ y ∈ X & α(y, g) = x}.”
Borel Equivalence Relations (2010, result): “This has been supported in various examples and practical investigations, such as [22], [49], as well as some partial results given in [32].”