Fréchet Borel ideals with Borel orthogonal
Volume 152 / 2018
Abstract
We study Borel ideals $I$ on $\mathbb {N}$ with the Fréchet property such that the orthogonal $I^\perp $ is also Borel (where $A\in I^\perp $ iff $A\cap B$ is finite for all $B\in I$, and $I$ is Fréchet if $I=I^{\perp \perp }$). Let $\mathcal {B}$ be the smallest collection of ideals on $\mathbb {N}$ containing the ideal of finite sets and closed under countable direct sums and orthogonal. All ideals in $\mathcal {B}$ are Fréchet, Borel and have Borel orthogonal. We show that $\mathcal {B}$ has exactly $\aleph _1$ non-isomorphic members. The family $\mathcal {B}$ can be characterized as the collection of all Borel ideals which are isomorphic to an ideal of the form $I_{\rm wf}{\upharpoonright }A$, where $I_{\rm wf}$ is the ideal on $\mathbb {N}^{ \lt \omega }$ generated by the well founded trees. Also, we show that $A\subseteq \mathbb {Q}$ is scattered iff ${\rm WO}(\mathbb {Q}){\upharpoonright } A$ is isomorphic to an ideal in $\mathcal {B}$, where ${\rm WO}(\mathbb {Q})$ is the ideal of well founded subsets of $\mathbb Q$. We use the ideals in $\mathcal {B}$ to construct $\aleph _1$ pairwise non-homeomorphic countable sequential spaces whose topology is analytic.
