% LaTeX2e
% Cohen forcing preserves being a gamma-set but not hurewicz property
% A. Miller  Sept 2012

\documentclass[12pt]{article}
\usepackage{amssymb}


\def\bb{{\mathfrak b}}
\def\cc{{\mathfrak c}}
\def\force{\forces}
\def\forces{{\;\Vdash}}
\def\ga{\gamma}
\def\la{\langle}
\def\name#1{\stackrel{\circ}{#1}}
\def\om{\omega}
\def\poset{{\mathbb P}}
\def\pr{\prime}
\def\proof{\par\noindent Proof\par\noindent}
\def\qed{\par\noindent QED\par\bigskip}
\def\ra{\rangle}
\def\res{\upharpoonright}
\def\rmand{\mbox{ and }}
\def\rmor{\mbox{ or }}
\def\sm{{\setminus}}
\def\st{\;:\;} % such that
\def\su{\subseteq}
\def\uu{{\mathcal U}}
\def\vv{{\mathcal V}}

\newtheorem{theorem}{Theorem}%[section]
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{example}[theorem]{Example}

\begin{document}

\begin{center}
{\large Cohen forcing preserves being a $\ga$-set 
\\
but not the Borel-Hurewicz property}
\end{center}

\begin{flushright}
Arnold W. Miller\\
Sept 2012
\end{flushright}

\def\address{\begin{flushleft}
Arnold W. Miller \\
miller@math.wisc.edu \\
http://www.math.wisc.edu/$\sim$miller\\
University of Wisconsin-Madison \\
Department of Mathematics, Van Vleck Hall \\
480 Lincoln Drive \\
Madison, Wisconsin 53706-1388 \\
\end{flushleft}}

Marion Scheepers proved that random real forcing preserves
being a $\ga$-set.
Boas Tsaban asked if the same is true for Cohen real forcing.

\begin{prop}
Suppose in the ground model $M$ that
$X\su 2^\om$ is a $\ga$-set and $\poset=2^{<\om}$
is Cohen real forcing.  Then for any $G$ $\poset$-generic
over $M$
$$M[G]\models X \mbox{ is a $\ga$-set.}$$
\end{prop}



\proof
Work in $M$.
Suppose $$p_0\forces \name{\uu} \mbox{ is an $\om$-cover of
$X$ with clopen sets and is downward closed.} $$
To simplify our notation assume that $p_0$ is the trivial condition
or replace $\poset$ by the conditions stronger than $p_0$.
For each $p\in\poset$ define 
$$\uu_p=\{C\;:\; \mu(C)<\frac1{2^{|p|+1}}\rmand
\exists q\leq p \;\;\; q\forces C\in \name{\uu}\}.$$
It is easy to check that each $\uu_p$ is an $\om$-cover of $X$.
Hence we may find $(C_p\in\uu_p\;:\; p\in\poset)$ a $\ga$-cover
of $X$.  Let
$f:\poset\to\poset$ be such
that $f(p)\leq p$ and $f(p)\forces C_p\in \name{\uu}$.

Let $G$ be $\poset$-generic over $M$ and define
$$\vv=\{C_p\;:\; f(p)\in G\}.$$
Then $\vv\su\uu$ is a $\ga$-cover of $X$.   Note that there
must be infinitely many $p$ with $f(p)\in G$ since no
$p$ can force that there are only finitely many.  The
measure condition on $\uu_p$ guarantees that $\vv$ is infinite.
\qed


Another question asked by Tsaban is whether it possible that
adding a Cohen real can destroy the  Hurewicz property in the
case of a totally imperfect set.  Scheepers
and Tall showed that adding a Cohen real destroys the property
that the ground model's Cantor set is Hurewicz.

\begin{prop}\label{two}
Suppose that $M\models X\su 2^\om$ is a Sierpinski set.
If $\poset$ is Cohen real forcing,  then for any
$G$ $\poset$-generic over $M$
$$M[G]\models X \mbox{ does not have the Hurewicz property.}$$
\end{prop}

\proof 
It is well-known that forcing with $\poset$ is equivalent to 
forcing with any nontrivial countable poset.  Here is the poset
we use:

$p\in \poset$ iff $p=(\vec{C_k}:k<n)$ for some $n$ where each
$\vec{C_k}=(C_{k,i}:i<n_k)$ is a finite sequence of
clopen sets in $2^\om$ with $\mu(\bigcup_{i<n_k}C_{k,i})<\frac1{2^k}$.
Then $p\leq q$ iff $n_p\geq n_q$ and $\vec{C_k}^p$ extends
$\vec{C_k}^q$ for $k<n_q$.  

Now let $G$ be $\poset$-generic over $M$ and in $M[G]$ define
$(\uu_n:n<\om)$ by 
$$\uu_k=\{(C_{k,i})^p\;:\; \exists p\in G \;\; k<(n)^p \rmand i<(n_k)^p\}.$$
It easy to check that each $\uu_k$ is a cover of $2^\om\cap M$ and
hence of $X$.  We claim that in $M[G]$ there does not exists $g:\om\to\om$
with the property that for every $x\in X$ there exists $N$ such that
$x\in \bigcup_{i<g(n)}C_{n,i}$ for all $n>N$.



Work in $M$.
Suppose for contradiction that there exists $p_0$ such that
$$p_0\forces \forall x\in X \;\exists N \;\forall n\geq N \;\;
\check{x}\in\bigcup_{i<\name{g}(n)}\name{C}_{n,i}.$$
For each $p\leq p_0$ and $N$ define
$$X(p,N)=\{x\in X\;:\; \forall n\geq N \;\; p\forces
\check{x}\in\bigcup_{i<\name{g}(n)}\name{C}_{n,i}\}.$$
Note that 
$$X=\bigcup\{X(p,N)\;:\; p\leq p_0 \rmand N<\om\}.$$
Since $\poset$ is countable there must exist $p\leq p_0$ and
$N$ for which $X(p,N)$ is uncountable and since $X$ is Sierpinski,
$X(p,N)$ has positive outer measure.  Note that if 
$q\leq p$ and $N^\pr\geq N$, then $X(q,N^\pr)\supseteq X(p,N)$.
Hence by extending $p$ and increasing $N$ if necessary we may 
suppose that 
\begin{enumerate}
\item $\mu^*(X(p,N))>\frac1{2^N}$,
\item $p\forces \name{g}(N)=\check{L}$, and
\item $N<n_p$ and the length of $(\vec{C}_N)^p$ is at least $L$.
\end{enumerate}
Since $\mu(\bigcup_{i<L}(C_{N,i})^p)<\frac1{2^N}<\mu^*(X(p,N))$,
we can choose $x\in X(p,N)$ with $x$ not in $\bigcup_{i<L}(C_{N,i})^p)$.
But this contradicts
$$p\forces \check{x}\in \bigcup_{i<\name{g}(N)}\name{C}_{N,i}
\rmand p\forces\name{g}(N)=\check{L}$$
\qed

Note that Sierpinski sets have the Hurewicz property with respect 
to Borel covers also.
Zdomskyy and Tsaban point out that 
Proposition \ref{two} directly contradicts Theorem 40
of Scheepers and Tall \cite{scheepers}.

\bigskip

Alan Dow asked ``Can adding one Cohen real lower the value of $\bb$?''

\medskip\noindent Proposition \ref{two} shows that this is possible.
Start with 
a model $M$ where $\bb=\om_2=\cc$ or any larger regular cardinal.
Then force with the measure algebra on $2^{\om_1}$.
In the model $M[H]$ the 
smallest unbounded set
is still $\om_2$ since the reals added by random real forcing
are bounded by ground model reals.  Let $G$ be $\poset$ generic over
$M[H]$.  Proposition \ref{two} gives us a sequence of covers 
$\uu_n=\{C_{n,m}:m<\om\}$ of the generic Sierpinski set $X_H$ determined
by $H$.  For each $x\in X_H$ let $f_x(n)$ be the least
$m$ with $x\in C_{n,m}$.  Then in $M[H][G]$ the set
$\{f_x:x\in X_H\}$ is unbounded in $\om^\om$.  Hence $\bb=\om_1$.

\begin{thebibliography}{99}

\bibitem{scheepers} Scheepers, Marion; Tall, Franklin D.; Lindelof
indestructibility, topological games and selection principles. 
Fund. Math. 210 (2010), no. 1, 1-46.

\end{thebibliography}

\bigskip\noindent
The following remark is due to Janusz Pawlikowski (email June 2013)

1.  any set that is null and Hurewicz is covered by a null $F_\sigma$ set,

2.  given models $M\subseteq N$:
if no real from $N$ is eventually different over $M$
(e.g., if the reals from $M$ are nonmeager in $N$),
then any null $F_\sigma$ set coded in $N$ is covered by a null
$G_\delta$ set coded in $M$,
so, if a nonnull set from $M$ becomes in $N$  null and Hurewicz,
then N adds an eventually different real over $M$,

3.  in particular, Cohen cannot force a Sierpinski set to keep the
Hurewicz property.






\end{document}