The theory of analytic,
Borelian and absolutely Baire spaces is applied to the theory of absolutely
Souslin and Borel Sets with respect to the class of all metrizable spaces.
Several intrinsic characterizations of absolutely Souslin and Borel sets are
given.
The basic idea is that if there is givem a class 𝒜 of respectable sets in a separable
theory, then the corresponding class 𝒞 in the associated nonseparable theory consisls
of all spaces P such that P = A∩G in βP where A ∈𝒜 and G is a Gδ set in βP. If
the elements of 𝒜 are characterized by existence of a complete structure of certain
type, then the elements of 𝒞 are characterized by the existence of a complete
bi-structure ⟨α,β⟩ where α is closely related to the structure defining 𝒜, and β is
closely related to the structure characterizing absolute Gδ spaces. This approach to
the nonseparable theory is discussed for analytic, Borelian and bi-analytic spaces.
The theory is applied to absolutely Borel and Souslin sets in the class of all
metrizable spaces.
