Uniform closed nowhere-dense covers for Borel meagre sections

Code an open section by a subset of the fixed cylinder list. If its code is Borel in $x$, then the relation $U_s\cap O_x\mathrm{\ne }\mathrm{\varnothing }$ is Borel: it is the countable disjunction over listed cylinders $U_t$ of the finite test $U_s\cap U_t\mathrm{\ne }\mathrm{\varnothing }$. Consequently the open set $O_x^{\mathrm{\prime }}=2^{\omega }\setminus \overline{O_x}=\bigcup \{U_s:U_s\cap O_x=\mathrm{\varnothing }\}$ has a Borel open code in $x$. Its boundary error $D_x=2^{\omega }\setminus (O_x\cup O_x^{\mathrm{\prime }})=\overline{O_x}\setminus O_x$ has a Borel closed code, is closed, and is nowhere dense: every open set meeting $\overline{O_x}$ meets $O_x$, hence no nonempty open set is contained in $D_x$.

We construct for every Borel $B\subseteq 2^{\omega }\times 2^{\omega }$ a Borel open-section code $O_x$ and Borel closed nowhere-dense-section codes $F_j(x)$ satisfying $B_x\mathrm{△}{O}_x\subseteq {\bigcup }_j{F}_j(x)\mathrm{.}$ For an open product set, list all basic product rectangles contained in it; their second-factor cylinders with first factor containing $x$ form the desired open section. Its error list is empty.

For $B={\bigcup }_i{B}_i$, use the induction data $(O_i,F_{i,j})$ and put $O={\bigcup }_i{O}_i$. If $y\in B_x\setminus O_x$, it lies in some $(B_i{)}_x\setminus (O_i{)}_x$. If $y\in O_x\setminus B_x$, it lies in some $(O_i{)}_x\setminus (B_i{)}_x$. Thus the symmetric difference is covered by the countable list $(F_{i,j}(x){)}_{i,j}$. The union open code and paired error list are Borel in $x$.

For $B^c$, let $O^{\mathrm{\prime }}$ be the interior of the complement of $O$ from step 1.1. Complementing both $B$ and $O$ preserves their symmetric difference, while $(2^{\omega }\setminus O_x)\setminus O_x^{\mathrm{\prime }}=D_x$. Hence $(B^c{)}_x\mathrm{△}{O}_x^{\mathrm{\prime }}\subseteq D_x\cup {\bigcup }_j{F}_j(x)\mathrm{.}$ Append the Borel closed nowhere-dense code $D_x$ to the old error list.

Every Borel set has a well-founded countable code built from open sets by complement and countable union. Recursion on that code using steps 1.2, 2.1, and 2.2 gives the asserted Borel data for $H$.

For a fixed $x$, both $H_x$ and its error set are meagre, so the open set $O_x$ is meagre. But no nonempty open subset of Cantor space is meagre. To see this directly, start with a cylinder inside such an open set and, against a given sequence of closed nowhere-dense sets, repeatedly choose the first strictly smaller subcylinder avoiding the next closed set. The nested finite words determine a point in the open set outside their union. Hence $O_x$ is empty and $H_x\subseteq {\bigcup }_j{F}_j(x)$. The recursion and least-cylinder fusion use fixed countable enumerations and no countable-choice selection of sectionwise covers. ∎