Finite products and the covering property

Products of ideals are a demanding field of study, laden with Ramsey-type considerations. In this section, we analyze a class of σ-ideals fairly frequent in practice, for which the product ideals possess a great degree of regularity as well as strong canonization properties. It is the class of σ-ideals σ-generated by coanalytic collection of compact sets, with the covering property:

Definition 9.1 A σ-ideal I has the covering property if every I-positive analytic set has a compact I-positive subset.

Since the σ-ideals σ-generated by closed sets have the continuous reading of names, and their quotient forcings are proper by Fact 6.1, in the context of this class of σ-ideals the covering property is equivalent to the bounding property of the quotient forcing by Fact 2.52.

Before the statement of the canonization theorem, recall the definition of the box product ideal from Fact 2.56.Let {Xi: i ∈ n} be Polish spaces with respective σ-ideals Ii on each. The box product Πi∈nIn is the collection of all Borel sets B ⊂ Πi∈nXi containing no subset of the form Πi∈nBn where for each i ∈ n, Bi ⊂ Xi is Borel and Ii-positive. Fact 2.56 has the following immediate corollary:

Corollary 9.2If the σ-ideals in the collection {Ii: i ∈ n} are σ-generated by a coanalytic collection of closed sets and have the covering property, then Πi∈n Ii is a σ-ideal.