A universal-meagre generic absorbs old nowhere-dense sets

Fix in $M$ a canonical enumeration $({\pi }_m{)}_{m<\omega }$ of the finite-prefix rearrangements of $2^{\omega }$: each is determined by a finite partial bijection between level-$n$ cylinders for some $n$, and there are only countably many such finite data, so the enumeration is definable without choice.

Perfect enlargement: let $C$ be any old nonempty closed nowhere-dense set. For every finite binary word $s$ with $C\cap [s]\mathrm{\ne }\mathrm{\varnothing }$, choose the first finite extension $t_s$ of $s$, in length-lexicographic order, for which $[t_s]\cap C=\mathrm{\varnothing }$. Such an extension exists by nowhere density. Put $K_s=\{t_s⌢y:(\mathrm{\forall }j)y(2j)=0\}$. This set is nonempty, closed, has no isolated points because arbitrarily late odd coordinates are free, and is nowhere dense because an arbitrarily late even coordinate can be set to $1$. Define $K=C\cup {\bigcup }_s{K}_s$. These are prescribed least choices and a set union, available in $M$ without Choice.

Grafting step: let $S$ be an old perfect nowhere-dense tree and let $(t,T)$ be a nontrivial condition. Below $1_{\mathrm{U}\mathrm{M}}$, first take the explicit nontrivial condition supplied by F1. Choose any $n>\mathrm{ht}(t)$ and any $\eta \in T\cap 2^n$; perfection of $T$ guarantees such a node. Enumerate the finite nonempty level $S\cap 2^n$ as $\{s_1,\dots ,s_k\}$. For each $i\le k$, put $C_i=\{\eta ⌢\tau :s_i⌢\tau \in S\}$, including all initial segments, and put $T^{\mathrm{\prime }\mathrm{\prime }}=T\cup {\bigcup }_{i\le k}{C}_i$. Thus all the level-$n$ sections of $S$ are grafted below the same node $\eta$; no comparison between the widths of $S$ and $T$ is needed. Every newly added node not already in $T$ has length greater than $n$, so $T^{\mathrm{\prime }\mathrm{\prime }}\cap 2^{\le n}=T\cap 2^{\le n}$ and in particular $T^{\mathrm{\prime }\mathrm{\prime }}\cap 2^{\le \mathrm{ht}(t)}=t$. Moreover $T\subseteq T^{\mathrm{\prime }\mathrm{\prime }}$, and $T^{\mathrm{\prime }\mathrm{\prime }}$ is perfect: nodes of $T$ keep their splitting extensions, while every node added from $C_i$ inherits splitting extensions from the section of the perfect tree $S$ below $s_i$. Hence $(t,T^{\mathrm{\prime }\mathrm{\prime }})\le (t,T)$ provided its body is nowhere dense, as checked below.

For completeness, $[U_G]$ is nowhere dense for an explicit dense-set reason. Given a word $s$ and a nontrivial condition $(t,T)$, choose an extension $v$ of $s$ with $[v]\cap [T]=\mathrm{\varnothing }$. Since $T$ is pruned binary, any node of $T$ has a branch by recursively taking the least available child; hence $v\mathrm{\notin }T$. Increase the recorded height to at least $\mathrm{\mid }v\mathrm{\mid }$. Every stronger condition omits $v$ permanently. These conditions are dense for each $s$, including below the weakest condition. The generic meets all these ground dense sets, so every cylinder has a subcylinder disjoint from $[U_G]$. The body is closed by F2, as required.

If a cylinder $[u]$ misses $C$, it meets no $K_s$ with $\mathrm{\mid }s\mathrm{\mid }\ge \mathrm{\mid }u\mathrm{\mid }$: intersecting cylinders would give $[s]\subseteq [u]$, contrary to $[s]\cap C\mathrm{\ne }\mathrm{\varnothing }$. It therefore meets only the finitely many $K_s$ indexed by shorter words. For any point outside $K$, first take such a cylinder around it and then avoid those finitely many closed sets, proving $K$ closed. Inside any cylinder first find a subcylinder missing $C$, then successively avoid the finitely many closed nowhere-dense $K_s$ meeting it; thus $K$ is nowhere dense. Every cylinder about a point of $C$ contains its corresponding nonempty $K_s$, disjoint from $C$, so no point of $C$ is isolated in $K$; points of the $K_s$ are not isolated either. Its prefix tree $S$ is consequently nonempty and perfect: any node meeting $K$ has two distinct extensions witnessed by two points of $K$ in its cylinder. F2 gives $[S]=K\supseteq C$. For $C=\mathrm{\varnothing }$, absorption is immediate and no enlargement is needed. Inclusion of the old prefix tree in $S$ implies inclusion of their bodies even for new branches in an extension.

The body of $T^{\mathrm{\prime }\mathrm{\prime }}$ is $[T^{\mathrm{\prime }\mathrm{\prime }}]=[T]\cup {\bigcup }_{i\le k}{\gamma }_i([S]\cap [s_i])$, where ${\gamma }_i(s_i⌢y)=\eta ⌢y$. Each displayed image is closed and has empty interior relative to the clopen cylinder $[\eta ]$, hence is closed nowhere dense in $2^{\omega }$. The union is finite, so together with the closed nowhere-dense set $[T]$ it is again closed nowhere dense. Thus $T^{\mathrm{\prime }\mathrm{\prime }}$ is a legitimate witness tree and $(t,T^{\mathrm{\prime }\mathrm{\prime }})$ is a condition below $(t,T)$.

Absorption below the condition: for each $i\le k$, choose a permutation of $2^n$ sending $\eta$ to $s_i$, and let ${\pi }_i$ be its induced finite-prefix rearrangement, which keeps the tail after the length-$n$ prefix unchanged. If $x\in [S]$, then uniquely $x=s_i⌢y$ for some $i$, while $\eta ⌢y\in [T^{\mathrm{\prime }\mathrm{\prime }}]$ by construction and ${\pi }_i(\eta ⌢y)=s_i⌢y=x$. Any generic filter containing $(t,T^{\mathrm{\prime }\mathrm{\prime }})$ has $[T^{\mathrm{\prime }\mathrm{\prime }}]\subseteq [U_G]$ by [F1]. Consequently $(t,T^{\mathrm{\prime }\mathrm{\prime }})$ forces $[S]\subseteq {\pi }_1([U_G])\cup \cdots \cup {\pi }_k([U_G])$. The ${\pi }_i$ occur in the fixed enumeration from step 1.1.

The conditions of the form $(t,T^{\mathrm{\prime }\mathrm{\prime }})$ of step 1.3 are dense below every condition: given $(t,T)$ and an old $S$, the grafting construction produces such a strengthening directly. For a nonempty old closed nowhere-dense set first apply the perfect-enlargement construction above to obtain its perfect enlargement; the empty set is automatic. Hence every condition forces that every old closed nowhere-dense set is contained in a finite subunion of the countable family $\{{\pi }_m([U_G]):m<\omega \}$.

In the extension, put $E={\bigcup }_{m<\omega }{\pi }_m([U_G])$. By step 4.1 and genericity, every old closed nowhere-dense set is contained in $E$; each ${\pi }_m([U_G])$ is closed nowhere dense because a homeomorphism preserves closedness and empty interior, so $E$ is a countable union of closed nowhere-dense sets and is meagre by [F3]. The code of $E$ is the generic tree together with the ground-model enumeration $({\pi }_m)$ of step 1.1, so $E$ is coded by the $\mathrm{U}\mathrm{M}$ generic.

Let $A\in M$ be meagre. By [F3] there are old closed nowhere-dense sets $[S_n]$ with $A\subseteq {\bigcup }_n[S_n]$, and by step 5.1 the union ${\bigcup }_n[S_n]$ is contained in $E$. Hence $A\subseteq E$: every old meagre set is contained in one meagre set coded by the generic.

The steps above establish both assertions of the Statement: the union of all old closed nowhere-dense sets is meagre, and every old meagre set is absorbed into the single coded meagre envelope $E$.