Borel master codes for null and meagre sets

Work in ZFC in Cantor space $\mathcal{C}=2^{\omega }$, the product of countably many copies of the discrete two-point space, with the product topology (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space) and the fair-coin measure $\mu$ of The fair-coin measure on Cantor space, so that $\mu$ is a Borel probability measure with $\mu ([s])=2^{-\mid s\mid }$ on the basic clopen cylinders $[s]=\{x\in \mathcal{C}:x\upharpoonright \mid s\mid =s\}$ (The Borel sigma-algebra of a topological space).

For this page define the Cantor-space null and meagre ideals by

${\mathcal{N}}_{\mathcal{C}}:=\{A\subseteq \mathcal{C}:\mathrm{\exists }B\in \mathcal{B}(\mathcal{C})(A\subseteq B\text{ and }\mu (B)=0)\},{\mathcal{M}}_{\mathcal{C}}:=\{A\subseteq \mathcal{C}:A\text{ is meagre in }\mathcal{C}\}\mathrm{.}$

Thus null membership means being contained in a Borel fair-coin null set; it does not assign $\mu (A)$ to a possibly non-Borel set $A$. These are the completed null ideal and the meagre ideal on $\mathcal{C}$. In ZFC, ${\mathcal{N}}_{\mathcal{C}}$ is closed under subsets and countable unions: choose a Borel null hull for each set in a countable family and take their Borel null union. The same closure holds for ${\mathcal{M}}_{\mathcal{C}}$ by flattening the countable nowhere-dense witnesses over $\mathbb{N}\times \mathbb{N}$; subsets preserve meagreness by definition. The unqualified symbols $\mathcal{N}$ and $\mathcal{M}$ in Add, cov, non and cof for null and meagre ideals denote the corresponding ideals on $\mathbb{R}$.

The fixed clopen basis. Fix once and for all a bijection $\mathbb{N}\to 2^{<\mathbb{N}}$ from the natural numbers onto the finite binary words, for definiteness the length-lexicographic one $\mathrm{\varnothing },0,1,00,01,10,11,000,\dots$, and write $U_k:=[s_k]$ for the $k$-th basic clopen set, so that $(U_k{)}_{k\in \mathbb{N}}$ enumerates the basic clopen sets of $\mathcal{C}$ and $\mu (U_k)\to 0$. Every open subset of $\mathcal{C}$ is a union of basic clopen sets, since the cylinders form a base of the topology.

Null master codes. A null master code is a function $f:\mathbb{N}\to \mathbb{N}$. Fix a second enumeration $(C_k{)}_{k\in \mathbb{N}}$ of all clopen subsets of $\mathcal{C}$, including the empty set, by coding finite unions of basic cylinders in length-lexicographic order. The null-code condition is

$\mu (C_{f(n)})\le 2^{-n}\text{for every }n\in \mathbb{N}\mathrm{.}$

The null set coded by $f$ is the limsup of the sequence of clopen sets that $f$ selects,

$N_f:={\bigcap }_{m\in \mathbb{N}}{\bigcup }_{n\ge m}{C}_{f(n)},$

which is Borel, and null: $\mu ({\bigcup }_{n\ge m}{C}_{f(n)})\le {\sum }_{n\ge m}{2}^{-n}=2^{1-m}$ by countable subadditivity and the geometric series, so continuity from above along the decreasing sequence of unions gives $\mu (N_f)={\lim }_m\mu ({\bigcup }_{n\ge m}{C}_{f(n)})=0$. Thus every null master code names a member of ${\mathcal{N}}_{\mathcal{C}}$, and the coded family is a family of null Borel sets.

Meagre master codes. A meagre master code is a function $f:\mathbb{N}\to \mathbb{N}$, read through a fixed bijection $\mathbb{N}\to \mathbb{N}\times \mathbb{N}$, such that for every $n\in \mathbb{N}$ the open set

$V_n(f):={\bigcup }_{m\in \mathbb{N}}{U}_{f(⟨n,m⟩)}$

is dense in $\mathcal{C}$. The meagre set coded by $f$ is the complement of the intersection of those dense open sets,

$M_f:=\mathcal{C}\setminus {\bigcap }_{n\in \mathbb{N}}{V}_n(f)={\bigcup }_{n\in \mathbb{N}}(\mathcal{C}\setminus V_n(f)),$

and it is meagre: each complement $\mathcal{C}\setminus V_n(f)$ is closed because $V_n(f)$ is open, and has empty interior because $V_n(f)$ is dense, so each complement is nowhere dense (Nowhere dense, meagre, residual, and comeagre subsets of a topological space) and $M_f$ is a countable union of closed nowhere dense sets, hence $M_f\in {\mathcal{M}}_{\mathcal{C}}$.

The slalom order. A slalom is a function $S$ with domain $\mathbb{N}$ and finite values $S(n)\subseteq \mathbb{N}$ (The cardinality $\mid A\mid$ of a finite set), subject to the summability condition ${\sum }_{n\in \mathbb{N}}\mid S(n)\mid 2^{-n}<\mathrm{\infty }$ (Series, partial sums, convergence and the sum, divergence, and the tail series). Slaloms are ordered by eventual inclusion,

$S{\subseteq }^{\ast }T:⟺S(n)\subseteq T(n)\text{ for all but finitely many }n\in \mathbb{N},$

and the space of slaloms with this preorder is written $(\mathbb{S},{\subseteq }^{\ast })$ below; it is the slalom space of the master-code construction. The order is reflexive and transitive, and it is one-sided: $S{\subseteq }^{\ast }T$ allows $S(n)⊋T(n)$ for finitely many $n$.

The two code families mirror each other. A null code fixes, at stage $n$, one finite union of cylinders of measure at most $2^{-n}$, and the coded set is the set of points falling into infinitely many stages; summability of the bounds is what makes the limsup null. A meagre code fixes, at stage $n$, a dense open set, and the coded set is the set of points falling outside at least one stage; density is what makes each of those complements nowhere dense. Nothing in the definitions requires the codes to be injective or the coded sets distinct: the master families $\{N_f\}$ and $\{M_f\}$ are used below for their cofinality in the respective ideals (Null and meagre master codes are cofinal), not for a bijective parametrisation of the ideals.

The name "master code" records that the coding is a presentation of the ideals just defined, not a parametrisation of their members. The symbols $\mathcal{N}$ and $\mathcal{M}$ in Add, cov, non and cof for null and meagre ideals refer to the real-line ideals; ${\mathcal{N}}_{\mathcal{C}}$ and ${\mathcal{M}}_{\mathcal{C}}$ here refer to Cantor space. The passage of the four cardinal invariants between these spaces is a separate matter needing separate measure and category maps (Transfer of null and meagre invariants between Cantor space and the line). Everything above is internal to $\mathcal{C}$ and consists of notation and elementary estimates; the constructions that give the master families their content — the uniform Borel section codes and the Tukey morphisms — are the lemmas that follow.

Both coding conditions are Borel conditions on the codes, in the sense needed for the parameterised arguments below. With the fixed enumerations of this definition each atomic condition, $f(n)=k$ or $U_j\cap U_k\mathrm{\ne }\mathrm{\varnothing }$, has a clopen truth set over the code space ${}^{\omega }\omega$, so the null-code condition and the density condition are countable Boolean combinations of clopen sets. Summability of ${\sum }_n\mid S(n)\mid 2^{-n}$ is Borel as well, being the union over $L\in \mathbb{N}$ of the conditions that every finite partial sum is at most $L$. No choice principle is used to code: the enumerations are fixed once and for all, and selecting a least witness index is a formula in the index.