Fréchet Borel ideals with Borel orthogonal

Abstract

We study Borel ideals I on N with the Fréchet property such that its orthogonal I ⊥ is also Borel (where A ∈ I ⊥ iff A ∩ B is finite for all B ∈ I and I is Fréchet if I = I ⊥⊥ ). Let B be the smallest collection of ideals on N containing the ideal of finite sets and closed under countable direct sums and orthogonal. All ideals in B are Fréchet, Borel and have Borel orthogonal. We show that B has exactly ℵ 1 non isomorphic members. The family B can be characterized as the collection of all Borel ideals which are isomorphic to an ideal of the form I wf ↾ A, where I wf is the ideal on N <ω generated by the wellfounded trees. Also, we show that A ⊆ Q is scattered iffis the ideal of well founded subsets of Q. We use the ideals in B to construct ℵ 1 pairwise non homeomorphic countable sequential spaces whose topology is analytic.

Ideals on countable sets: a survey with questions

“…The ideal I wf is a complete co-analytic set, while the ideal I do is easily seen to be F σδ , it is not countably generated and it is not selective (see [10, Example 2]). In [27] was constructed a family B of ℵ 1 non isomorphic Fréchet ideals such that both I and I ⊥ are Borel. In fact, every ideal in B is F σδ .…”

“…Theorem 9.15. (Guevara-Uzcátegui [27]) Let I be an analytic selective ideal on N and A ⊆ N. The following statements are equivalent:…”

“…A linear order (L, <) is said to be scattered, if it does not contain a order-isomorphic copy of Q. Theorem 9.17. (Guevara-Uzcátegui [27]) For every A ⊆ Q, the following statements are equivalent:…”

Selective separability on spaces with an analytic topology

“…Recall that a filter F is Fréchet if for all A ∈ F + there is B ⊆ A such that every infinite subset of B belongs to F + . In [10] were constructed ℵ 1 non homeomorphic spaces of the form Seq(F) with F a Fréchet analytic filter.…”

“…In [10] were constructed ℵ 1 non homeomorphic spaces of the form Seq(F) with F a Fréchet analytic filter. Now we address the question of when Seq(F) is q + .…”