FRECHET BOREL IDEALS WITH BOREL ORTHOGONAL

We study Borel ideals I on N with the Frechet property such that the orthogonal I-perpendicular to is also Borel (where A is an element of I-perpendicular to iff A boolean AND B is finite for all B is an element of I, and I is Frechet if I = I-perpendicular to perpendicular to). 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 13 are Frechet, Borel and have Borel orthogonal. We show that B has exactly N-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 (sic)A, where I-wf is the ideal on N-<omega generated by the well founded trees. Also, we show that A subset of Q is scattered if WO(Q)(sic)A is isomorphic to an ideal in B, where WO(Q) is the ideal of well founded subsets of Q. We use the ideals in B to construct N-1 pairwise non-homeomorphic countable sequential spaces whose topology is analytic.