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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.