We present a method for showing that a given Π03 subset of a Polish space is in fact Π03-complete. This is motivated by some questions from V. Nestoridis about the sequential spaces ℓp and more generally about families of F-spaces (Xi)i∈(I,⪯) that form ⊆-chains, where ⪯ is a linear ordering.
The intersection ∩p>aℓp is known to be a Π03 subset of ℓq for all a,q with 0≤a<q<∞ (Nestoridis). We show that it is in fact a Π03-complete set. It turns out the proof can be generalized to the context of Polish spaces with no additional structure like linearity. This gives a method for showing Π03-completeness and in fact there are strong indications that it also gives a characterization of the latter property.