P-ideals, PID, the pseudointersection number, and S-spaces

For subsets of a set $A$, write $x{\subseteq }^{\ast }y$ when $x\setminus y$ is finite, and write $x\perp y$ when $x\cap y$ is finite. If $\mathcal{F}\subseteq \mathcal{P}(A)$, then

${\mathcal{F}}^{\perp }=\{x\subseteq A:(\mathrm{\forall }y\in \mathcal{F})x\perp y\}\mathrm{.}$

An ideal of countable subsets of $A$ is a family $\mathcal{I}\subseteq [A{]}^{\le \omega }$ that contains every finite subset of $A$, is downward closed, and is closed under finite unions. It is a P-ideal if for every sequence $⟨a_n:n<\omega ⟩$ in $\mathcal{I}$ there is $b\in \mathcal{I}$ such that $a_n{\subseteq }^{\ast }b$ for every $n$. A set $X\subseteq A$ is orthogonal to $\mathcal{I}$ when $X\in {\mathcal{I}}^{\perp }$, that is, $X\cap a$ is finite for every $a\in \mathcal{I}$.

The P-ideal dichotomy (PID) says that for every such P-ideal $\mathcal{I}$ on every set $A$, at least one of the following holds:

1. there is an uncountable $B\subseteq A$ such that $[B{]}^{\le \omega }\subseteq \mathcal{I}$;
2. there are sets $X_n\subseteq A$ for $n<\omega$ with $A={\bigcup }_{n<\omega }{X}_n$ and each $X_n\perp \mathcal{I}$.

A family $\mathcal{A}\subseteq [\omega {]}^{\omega }$ has the strong finite intersection property if $\bigcap \mathcal{F}$ is infinite for every finite $\mathcal{F}\subseteq \mathcal{A}$ (including $\mathcal{F}=\mathrm{\varnothing }$, whose intersection is $\omega$). An infinite $b\subseteq \omega$ is a pseudointersection of $\mathcal{A}$ when $b{\subseteq }^{\ast }a$ for every $a\in \mathcal{A}$. The pseudointersection number $\mathfrak{p}$ is the least cardinality of a strong-finite-intersection family in $[\omega {]}^{\omega }$ with no infinite pseudointersection. This cardinal-invariant clause is read in ZFC: AC well-orders the possible witness sizes and supplies the standard existence argument for a witnessing centered family.

A topological space is hereditarily separable (respectively, hereditarily Lindelöf) if every one of its subspaces is separable (respectively, Lindelöf). An S-space is a regular Hausdorff, hereditarily separable, non-Lindelöf space. Here regularity and Hausdorffness are both stated because this library's word “regular” does not by itself carry a separation axiom. The empty and singleton spaces are Lindelöf, so neither is an S-space. These are predicates and cardinal definitions only; no individual witness is selected in this item, but the existence and well-defined cardinal value of $\mathfrak{p}$ use ambient AC as just stated.