Luzin sets, stick, and almost-disjoint guessing at omega one

Work in ZFC, with the choice axiom The Axiom of Choice. Throughout this small-space construction, $\kappa ={\omega }_1$ and $E$ is the set of nonzero countable limit ordinals, equivalently $E_{\omega }^{{\omega }_1}$. This changes the preceding continuum-sized construction's notation. The facts that ordinals below ${\omega }_1$ are countable and countable subsets of ${\omega }_1$ are bounded are ${\omega }_1$ is uncountable, every ordinal below it is at most countable, it is a cardinal and a limit ordinal, and its existence is a theorem of ZF and Assuming countable choice: every at most countable subset of ${\omega }_1$ is bounded below ${\omega }_1$, so no at most countable subset of ${\omega }_1$ is cofinal in it, and a supremum of at most countably many at most countable ordinals is at most countable; their countable-choice hypothesis follows by restricting AC. Every nonzero countable limit has a cofinal increasing sequence: enumerate the ordinal, and successively choose an ordinal above the previous choice and the next enumerated value. Finite sets are bounded in a limit ordinal, so its cofinality is exactly $\omega$. A club is an unbounded subset containing its nonzero limit points below $\kappa$; a stationary set meets every club.

1. In Baire space $\mathcal{N}={\omega }^{\omega }$ of Baire sequence space ${\mathbb{N}}^{\mathbb{N}}$ and its cylinder topology, write ${\omega }^{<\omega }$ for the finite strings and $N_t=\{x:t\subseteq x\}$ for a cylinder. The empty string gives $N_{\mathrm{\varnothing }}=\mathcal{N}$. A set is nowhere dense when its closure has empty interior, and meager when it is a countable union of nowhere dense sets. A closed $F\subseteq \mathcal{N}$ is nowhere dense exactly when every cylinder has a subcylinder disjoint from $F$: an open complement meeting each cylinder gives such a subcylinder, and a cylinder in the interior would contradict the subcylinder condition. A classical Luzin set in $\mathcal{N}$, or in $\mathbb{R}$ with its usual topology, is an uncountable set meeting every meager set in an at most countable set. A set $L\subseteq \mathcal{N}$ of size ${\mathrm{\aleph }}_1$ has the ${\omega }_1$-Luzin cylinder property if for every uncountable $B\subseteq L$ there is $t\in {\omega }^{<\omega }$ such that every finite extension $u\supseteq t$ is extended by an element of $B$. This says $B$ is dense in $N_t$; it is a separate definition from classical Luzin.

2. Stick at ${\omega }_1$ asserts a sequence $(s_{\xi }{)}_{\xi <\kappa }$ of countably infinite subsets of $\kappa$ such that every uncountable $X\subseteq \kappa$ contains some $s_{\xi }$. Repetitions in the sequence are allowed. Empty and countable targets carry no requirement.

3. Let $\mathcal{P}$ be a partition of $E$ into stationary sets. An AD guessing array for $\mathcal{P}$ is $(A_{\alpha }^i{)}_{\alpha \in E,i<\omega }$ with $A_{\alpha }^i\subseteq \alpha$ cofinal, with $A_{\alpha }^i\cap A_{\alpha }^{i^{\mathrm{\prime }}}=\mathrm{\varnothing }$ for $i\mathrm{\ne }{i}^{\mathrm{\prime }}$, and with $A_{\alpha }^i\cap A_{\beta }^j$ bounded in $\alpha$ whenever $\alpha <\beta$ lie in $E$. Its guessing requirement is: for every $1\le r<\omega$, every list of uncountable sets $X_0,\dots ,X_{r-1}\subseteq \kappa$, and every $S\in \mathcal{P}$, the set

   $\{\alpha \in S:(\mathrm{\forall }i<\omega )(\mathrm{\forall }j<r)\sup (A_{\alpha }^i\cap X_j)=\alpha \}$

   is stationary. Lists may repeat targets. This is ${\mathrm{♣}}_{\mathrm{A}\mathrm{D}}(\mathcal{P},\omega ,<\omega )$, with bounded intersections, which need not be finite.

4. The strong diagonal AD property requires disjoint cofinal members in every row and finite intersections $A_{\alpha }^i\cap A_{\beta }^j$ for distinct row indices $\alpha \mathrm{\ne }\beta$. For every sequence $(X_{\nu }{)}_{\nu <\kappa }$ of uncountable subsets of $\kappa$ and every $S\in \mathcal{P}$, it requires stationarity of

   $\{\alpha \in S:(\mathrm{\forall }i<\omega )(\mathrm{\forall }\nu <\alpha )\sup (A_{\alpha }^i\cap X_{\nu })=\alpha \}\mathrm{.}$

   This is the ${\omega }_1$ instance of ${\mathrm{♣}}_{{\mathrm{A}\mathrm{D}}^{\ast }}(\mathcal{P},\omega ,{\omega }_1)$. It implies clause 3: pad a finite list by $\kappa$, and intersect the stationary guessing set with the club tail above its length. Finite intersections are bounded in the nonzero limit indices.

5. A single-ladder two-target AD system consists of pairwise disjoint stationary sets $(S_n{)}_{1\le n<\omega }$ contained in $E$, with union $S$, and sets $A_{\alpha }\subseteq \alpha$ cofinal for $\alpha \in S$. For $\alpha <\beta$ in $S$, require $A_{\alpha }\cap A_{\beta }$ bounded in $\alpha$. For every two uncountable $X_0,X_1\subseteq \kappa$ and each $n\ge 1$, require stationarily many $\alpha \in S_n$ with $\sup (A_{\alpha }\cap X_0)=\sup (A_{\alpha }\cap X_1)=\alpha$. An array in clause 3 on such a partition supplies this by taking its zeroth member, $A_{\alpha }=A_{\alpha }^0$. No order-type-$\omega$ requirement is imposed on these ladders.

These definitions assert no ZFC existence. Zero and successors are excluded as final ladder indices. Ordinary diamond is the subset-guessing principle of Diamond on ω1, and Ostaszewski $\mathrm{♣}$ is the uncountable-target containment principle of The Ostaszewski club principle. Ordinary club guessing tests club targets; this differs from testing all uncountable targets. Parameterized $\mathrm{♢}(\mathfrak{b})$ is also a different principle, and is not an abbreviation for ordinary diamond here.