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\newcommand{\Fcube}{{\cal F}_{\rm cube}}
\newcommand{\Fpr}{{\cal F}_{\rm prism}}

\newcommand{\Ccube}{{\cal C}_{\rm cube}}
\newcommand{\Cpr}{{\cal C}_{\rm prism}}

\newcommand{\IPSM}{{\sc Sacks Model}}

\newcommand{\cov}{{\rm cov}}

\newcommand{\psmPrGame}{{\rm CPA$_{\rm prism}^{\rm game}$}}
\newcommand{\psmCuGame}{{\rm CPA$_{\rm cube}^{\rm game}$}}

\newcommand{\dec}{{\rm DEC}}
\newcommand{\IntTh}{{\rm IntTh}}
\newcommand{\Equi}{\Longleftrightarrow}
\newcommand{\SoIC}{{s_0^{\rm ic}}}
\newcommand{\SoST}{{s_0^{\rm pc}}}

\newcommand{\psm}{{\rm CPA}}
\newcommand{\psmC}{\mbox{{\rm CPA$_{\rm cube}$}}}
\newcommand{\psmP}{\mbox{{\rm CPA$_{\rm prism}$}}}
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\newcommand{\psmPrPLUS}{{\rm CPA$_{\rm prism}^+$}}
\newcommand{\cpa}{{\rm CPA}}

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% My Commands (Andres)

\newcommand{\Baire}{{\omega^{\omega}}}
\newcommand{\Ramsey}{{[\omega]^{\omega}}}
\newcommand{\Power}{{2^{\omega}}}
\newcommand{\almost}{{\subset^*}}
\newcommand{\Fin}{{[\omega]^{<\omega}}}
\newcommand{\I}{{\mathcal{I}}}
\newcommand{\Uncount}{{\omega_1}}
\newcommand{\gamecube}{{\text{GAME}_{\rm cube}(\X)}}
\newcommand{\gameprism}{{\text{GAME}_{\rm prism}(\X)}}
\newcommand{\projkeep}{{\Phi_{\rm prism}(\alpha)}}
\newcommand{\Z}{{\mathcal{Z}}}

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\def\inter{\mathop{\rm int}}
\def\cof{\mathop{\rm cof}}
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\newtheorem{corollary}{Corollary}
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\begin{document}

\title {\LARGE  A crowded $Q$-point under \psmPrGame}

\author{Andres Mill\'an%
\thanks{AMS subject classification numbers: Primary 54D80; Secondary 03E35.
Keywords and phrases: ultrafilter, crowded, $Q$-point, Covering Property Axiom \cpa.
This work is a part of author's Ph.D. thesis written at
the West Virginia University under the supervision of
Krzysztof Ciesielski. The author wishes to thank professor Ciesielski for his guidance,
patience and encouragement.}}

\maketitle

\begin{abstract}

In this note we prove that the version \psmPrGame \; of
the Covering Property Axiom, which holds in the iterated Sacks model,
implies that there exists an $\omega_1$-generated crowded ultrafilter
on $\Q$ which is also a $Q$-point. Since no crowded ultrafilter can be
a $P$-point this constitutes an interesting example of a $Q$-point
which is not a $P$-point.

\end{abstract}



\section{Introduction}

We will use standard set theoretic notation as in \cite{SetTheory}.
Let $\U$ be a non-principal ultrafilter on a countable set $X$.
Then, $\U$ is a \emph{$P$-point} if for every partition
$\P$ of $X$ either $\U\cap{\P}\neq{\emptyset}$ or
there exists an $X\in{\U}$ such that $X\cap{P}$ is finite for each
$P\in{\P}$. $\U$ is called a \emph{$Q$-point} if for every partition
$\P$ of $X$  into finite pieces there exists an $X\in{\U}$ such that
$|X\cap{P}|\leq{1}$ for each $P\in{\P}$.
Given a non-principal ultrafilter $\U$ on $X$ we say that $\B\subset{\U}$
is a basis for $\U$ if for every $U\in{\U}$ there exists a $B\in{\B}$ such
that $B\subset{U}$. Then, we can define the \emph{character} of $\U$ as
$\chi(\U)=\text{min}\{|\B| \colon \B \text{ is a basis for } \U\}$. We say that
$\U$ is $\kappa$-generated if $\chi(\U)=\kappa$.

Consider $\Q$ with the subspace topology induced by the usual topology
on $\real$ and denote by $\perf{(\Q)}$ the family of its perfect subsets.
A non-principal filter $\U$ on $\Q$ is \emph{crowded} if the family
$\perf{(\Q)}\cap{\U}$ forms a basis for $\U$. The crowded ultrafilters have
been studied in connection with the remainder of the Stone-\v{C}ech
compactification of $\Q$ and their existence follows from the Continuum
Hypothesis,
Martin's Axiom for countable posets \cite{crowded1}, or from
the equality
$\mathfrak{b}=\continuum$ \cite{crowded2}.

In \cite{crowded3} Ciesielski and Pawlikowski showed that a version of their
Covering Property Axiom called \psmPrGame, which holds in the iterated
Sacks model, implies that there exists an $\omega_1$-generated crowded
ultrafilter
on $\Q$ and they noted that no crowded ultrafilter can be a $P$-point.
This result is interesting because \psm \; implies $\mathfrak{b}<\continuum$.

The main result of this paper is that \psmPrGame \; implies the existence
of an $\omega_1$-generated crowded ultrafilter on $\Q$ which is also a
$Q$-point
\footnote{Recently the author has proven that \psmPrGame \; implies that
there is also a crowded $Q$-point of character $\continuum$.}.
Notice that this contradicts the remark by Ciesielski and Pawlikowski in
\cite[page 49]{crowded3} that crowded ultrafilters cannot be $Q$-points.

It is a result of A.Miller \cite{qpoint} that there are no
$Q$-points in Laver's \cite{Borel} model for Borel's Conjecture.
Since the equality  $\mathfrak{b}=\continuum$ holds in Laver's model,
it is consistent with ZFC that no crowded ultrafilter on $\Q$ is a $Q$-point.


\section{Preliminaries on \psmCuGame \; and \psmPrGame}

\subsection{Cubes and Prisms.}

The framework of \cpa \; rests on the concepts of \emph{cube} and
\emph{prism}. If $\Cantor$ denotes the space $2^{\omega}$ with its usual
product topology and $\X$ is a Polish space then we define
$$\perf{(\X)}=\{C\subset{\X} \colon C \text{ is homeomorphic to }
\Cantor\}.$$
A \emph{perfect cube} in $\Cantor^{\omega}$ is any set
$C=\prod_{i<\omega}C_i$ where
$C_i\in{\perf{(\Cantor)}}$ for every $i<\omega$.
If $\X$ is a Polish space, then a \emph{cube} in $\X$ is a pair $\la{f,P}\ra$
where $f \colon C \to \X$ is a continuous injection and $P=f[C]$ for some
perfect cube $C$.
The following theorem is one of the principal tools for using \psm,
and it is a refinement of a theorem proved independently
by H.G. Eggleston and M.L. Brodski\u{\i}.

\prop{prop:teorema1}{{(\rm K.Ciesielski, J.Pawlikowski \cite[claim
1.1.5]{book})}
Consider $\Cantor^{\omega}$ with its usual topology and its usual product
measure.
If $G$ is a Borel subset of $\Cantor^{\omega}$ which
is either of second category or of positive measure then
$G$ contains a perfect cube.}


The notion of \emph{prism} is a generalization of that of a cube.
If $\alpha<\omega_1$ is a non-zero countable ordinal let $\projkeep$
be the set of all functions $f \colon \Cantor^{\alpha} \to \Cantor^{\alpha}$
with the property that
$$f(x)\restriction{\xi}=f(y)\restriction{\xi}
\Leftrightarrow{x\restriction{\xi}=y\restriction{\xi}}
\text{\qquad for all  $\xi<\alpha$ and $x,y\in{\Cantor^{\alpha}}$}.$$
Then we define
$\mathPerf_{\alpha}=\{\text{range}(f) \colon f\in{\projkeep}\}$
and
$\mathPerf_{\omega_1}=\bigcup_{0<\alpha<\omega_1}\mathPerf_{\alpha}.$
%
The elements of $\mathPerf_{\omega_1}$ are called the \emph{iterated
perfect sets}.
If $\X$ is a Polish space, then a \emph{prism} on $X$ is a pair $\la{f,P}\ra$
where $f \colon E \to \X$ is injective and continuous,
$E\in{\mathPerf_{\omega_1}}$, and $P=f[E]$. 

It is also inmediate to observe that if the pair $\la{f,P}\ra$ and 
$f \colon E \to P$ and $E\in{\mathPerf_{\alpha}}$ then, we can assume
that $f$ is defined on the entire $\Cantor^{\alpha}$. 


It is important to note that the previous definitions imply that perfect
cubes are, in particular, iterated perfect sets and therefore, that cubes are
prisms. On the other hand, if $\la{g,P}\ra$ is a prism,
where $g\colon E\to P$ and $E\in\mathPerf_{\alpha}$,
then there exists an $f\in{\projkeep}$ with $E=\text{range}(f)$.
In particular, $h=g\circ f\colon\Cantor^\alpha\to P$
is a continuous injection and
the pair $\la{h,P}\ra$ is a cube. Thus, any prism can be thought
as a cube with a different coordinate system imposed on it.

%Since we will need to consider singletons in $X$ we will
%include these in our definition of cube and prism.


\subsection{Subcubes and Subprisms.}

If $\la{f,P}\ra$ is a cube, then we say that $Q$ is its subcube provided
there exists a perfect cube $C\subset{\text{dom}(f)}$ such that $Q=f[C]$.
Subprisms are defined similarly but replacing  the perfect cube $C$ by
an iterated perfect set $E$.
Since in the games defined below we will need to consider singletons
in the same position as cubes (or prism) as defined above,
in what follows {\em singletons will be considered as cubes and prisms.}
If $P$ is a singleton in $\X$ then its only
subcube is $P$ itself.


\subsection{Games and Strategies.}

For a Polish space $\X$ consider the following game $\gamecube$ of length
$\omega_1$ played by two players, Player I and Player II. At each stage
$\xi<\omega_1$ of the game Player I can play an arbitrary cube $P_{\xi}$ in $\X$
(i.e., $P_{\xi}$ either belongs to $\perf(\X)$ or is a singleton in $\X$)
and Player II must respond by playing
a subcube $Q_{\xi}$ of $P_{\xi}$.
The game $\la{\la{P_{\xi},Q_{\xi}}\ra \colon \xi<\omega_1}\ra$ is won by
Player I provided $$\X=\bigcup_{\xi<\omega_1}Q_{\xi};$$
otherwise Player II wins.

A strategy for Player II is any function $S$ such that
$S(\la{\la{P_{\eta},Q_{\eta}}\ra \colon \eta<\xi}\ra, P_{\xi})$
is a subcube of $P_{\xi}$ for every partial game
$\la{\la{P_{\eta},Q_{\eta}}\ra \colon \eta<\xi}\ra$.
We say that a game $\la{\la{P_{\xi},Q_{\xi}}\ra \colon \xi<\omega_1}\ra$
is played according to a strategy $S$ for Player II provided
$Q_{\xi}=S(\la{\la{P_{\eta},Q_{\eta}}\ra \colon \eta<\xi}\ra, P_{\xi})$
for every $\xi<\omega_1$. A strategy $S$ for Player II is a \emph{winning
strategy}
provided Player II wins any game played according the strategy $S$.
The corresponding notions of games, strategies etc. for prisms are defined
in a similar way.

\subsection{The Axioms.}


The following principles capture the combinatorial
core of the iterated Sacks model.

\bigskip

\noindent
\psmCuGame : $\continuum=\omega_2$ and for any Polish space $\X$ Player II
has no winning strategy in the game $\gamecube$.

\bigskip

\noindent
\psmPrGame : $\continuum=\omega_2$ and for any Polish space $\X$ Player II
has no winning strategy in the game $\gameprism$.

\bigskip

\noindent
These axioms are consequences of a more general principle, similar in spirit,
called  \cpa \; \cite{book}. Their importance comes from the following theorem.

\prop{prop:teorema2}{{\rm (K.Ciesielski, J.Pawlikowski
\cite{crowded3,book})} \cpa \; holds in the iterated
perfect set model. In particular, \psm \; is consistent with ZFC set theory.}



\section{An $\omega_1$-generated crowded $Q$-point on $\Q$}

If the set $X=[\omega]^{<\omega}\setminus{\{\emptyset\}}$
has the discrete topology then the product space
$\X=X^{\omega}$ is a Polish space and the sets
$U_{\la{n,a}\ra}=\{x\in{\X} \colon x(n)=a\}$,
where $a\in{[\omega]^{<\omega}}$ and $n<\omega$, constitutes
a subbasis for the product topology.
Consider the set
$$\P=\{x\in{\X} \colon \mbox{ $\{x(k) \colon k<\omega\}$
is a partition of $\omega$}\}.$$

\noindent
It is important to know that

\begin{itemize}
\item{$\P$ is a $G_{\delta}$ subset of $\X$. Therefore, $\P$ is a
Polish space with the relative topology inherited from $\X$.}
\end{itemize}


\lem{lem:lema1}{Let $P$ be a prism in $\P$ and let
$\{A_n \colon n<\omega\}\subset{[\Q]^{\omega}}$ be arbitrary. 
Then, there exist a subprism $Q$ of $P$ and $B\in{[\Q]^{\omega}}$ such that
$|B\cap{A_n}|=\omega$ for every $n<\omega$ and
$|x(k)\cap{B}|\leq{1}$ for every $x\in{Q}$ and $k<\omega$.
Moreover, if $P$ is a cube then, $Q$ is a cube as well.}
%
\proof
Since $|\Q|=\omega$ we can suppose that 
$\{A_n \colon n<\omega\}\subset{[\omega]^{\omega}}$.
Let $\la R_n \colon n<\omega\ra$ be an enumeration of 
$\{A_n \colon n<\omega\}$ where each set appears infinitely often.

Case (a): If $P=\{z\}$ then, define a sequence $\la{b_n\in{\omega} \colon n<\omega}\ra$
inductively such that 
$b_n\in{R_n\setminus{\bigcup\{z(k) \colon k<\omega\ \&\ z(k)
\cap{\{b_0,\dots,b_{n-1}\}\neq{\emptyset}}\}}}$ for every $n<\omega$.
It is easy to see that $B=\{b_n \colon n<\omega\}$ works.

Case (b): If $P\in{\perf(\P)}$, let $f$ be a witness function for $P$. 
By our remarks in section 2, we can assume that $f$ acts from $\Cantor^\alpha$ onto $P$.
Thus, $P$ is a cube. It is enough to find its subcube with the desired
properties.

Let $\mu$ be the standard product probability measure on
$\Cantor^\alpha$.
%

%

We construct, by induction on $n<\omega$, a sequence
$\la{K_n\colon n<\omega}\ra$ of open subsets of $\Cantor^\alpha$ and two sequences, 
$\la{b_n\in{R_n}} \colon n<\omega\ra$
and $\la{B_n\in{[\omega]^{<\omega}}} \colon n<\omega\ra$,
such that for every $n<\omega$:
%
%
\begin{itemize}
\item[(i)] $b_n>\max\left(\{b_i \colon i<n\}\cup{\bigcup_{j<n}B_j}\right)$,
\item[(ii)] $\mu{(K_n)}\geq 1-2^{-(n+2)}$, and
\item[(iii)] $f(h)(k)\subseteq B_n$ for every $h\in K_n$ and $k<\omega$ for
which $b_n\in f(h)(k)$.
\end{itemize}

If this construction is possible put $B=\{b_n \colon n<\omega\}$.
Then, clearly  $|B\cap{A_n}|=\omega$. 
Condition (ii) implies
that $\mu\left(\bigcap_{n<\omega}K_n\right)\geq{\frac{1}{2}}$.
Hence, by Proposition~\ref{prop:teorema1}, there exists a perfect cube
$C\subseteq{\bigcap_{n<\omega}K_n}$. Then $Q=f[C]$ is a subcube of $P$
and the pair $\la{Q,B}\ra$ is as required.
To see this, it is enough to show that $|z(k)\cap B|\leq 1$
for every $z\in{Q}$ and $k<\omega$.
Let $z=f(h)$ for some $h\in C$.
By conditions (i) and (iii), for every $b_j\in z(k)=f(h)(k)$ and $n>j$
we have that $b_n\notin z(k)$.
Therefore, no two elements of $B$ are in the same $z(k)$ or, in
other words, $|z(k)\cap{B}|\leq{1}$ for every $k<\omega$.

Next, we show that the inductive construction is possible.
Let $n<\omega$ be such that the appropriate $b_i$, $K_i$, and $B_i$ are already
constructed for every $i<n$. We will construct $b_n$, $K_n$, and  $B_n$
satisfying (i)--(iii).
We pick an $b_n$ as an arbitrary element of $R_n$ satisfying (i).
If $L=\{a\in[\omega]^{<\omega}\colon b_n\in a\}$
then, $\left\{f^{-1}\left(U_{\la m,a\ra}\right)\colon 
\la m,a\ra\in\omega\times L\right\}$
is a partition of $\Cantor^\alpha$ into clopen sets.
Thus, we can find a finite set $S\subseteq \omega\times L$
such that
$K_n=\bigcup\left\{f^{-1}\left(U_{\la m,a\ra}\right)\colon 
\la m,a\ra\in S\right\}$
satisfies condition (ii).
Let
$B_n=\bigcup\{a\colon \la m,a\ra\in S\mbox{ for some }m<\omega\}$.
Then clearly, $B_n$ is finite. To see that it satisfies (iii)
take an $h\in K_n$.
Then $f(h)\in U_{\la m,a\ra}$ for some $\la m,a\ra\in S$.
Let $k<\omega$ be such that $b_n\in f(h)(k)$.
Since we have also $b_n\in a=f(h)(m)$
we conclude that $k=m$. So, $f(h)(k)=f(h)(m)=a\subseteq B_n$.
\qed

%\noindent
Fix a $p\in{\real\setminus{\Q}}$. For $\D\subset{[\Q]^{\omega}}$ let
$F(\D)=F(p,\D)$ be the filter generated by the family $\D\cup{\{I_n \colon
n<\omega\}}$,
where $I_n=[p-2^{-n},p+2^{-n}]\cap{\Q}$.



\lem{lem:lema2}{{\rm (K.Ciesielski, J.Pawlikowski \cite[lemma 4.23]{crowded3})}
Let $\D\subset{\perf{(\Q)}}$ be a countable family such that $F(\D)$
is crowded. Then, for every prism $P$ in $[\Q]^{\omega}$ there exists
a subprism $Q$ of $P$ and a $Z\in{\perf{(\Q)}}$ such that $F(\D\cup{\{Z\}})$
is crowded and either
\begin{itemize}
\item[(i)]{$Z\cap{x}=\emptyset$ for every $x\in{Q}$, or else}
\item[(ii)]{$Z\subset{x}$ for every $x\in{Q}$.}
\end{itemize}}

%\noindent
We will need also the following easy fact.

\lem{lem:lema3}{{\rm (K.Ciesielski, J.Pawlikowski \cite[Fact 4.21]{crowded3})}
Every non-scattered set $B\subset{\Q}$ contains a subset from $\perf{(\Q)}$.}

\lem{lem:lema4}{Let $\D\subset{\perf{(\Q)}}$ be a countable family such
that $F(\D)$ is crowded and let $P$ be prism in $\P$ 
then there exists a subprism $Q$ of $P$ and $Z\in{\perf{(\Q)}}$ 
such that $F(\D\cup{\{Z\}})$ is crowded
and $|Z\cap{x(k)}|\leq{1}$ for every $x\in{Q}$. }

\proof Observe that since $F(\D)$ is crowded it is possible to find a
sequence $\la{D_n\in{\perf{(\Q)}} \colon n<\omega}\ra$ coinitial in $F(\D)$
such that $D_{n+1}\subset{D_n}\subset{I_n}$ for every $n<\omega$.
Note that

\begin{itemize}
\item there are sequences
$\la{J_k\colon k<\omega}\ra$ of pairwise disjoint intervals in $\rational$ and
$\la{S_k\subset J_k \colon k<\omega}\ra$ of perfect subsets of $\rational$
such that if $S=\bigcup_{k<\omega}S_k$ then for every $D\in{F(\D)}$ there
exists
an $n<\omega$ such that $S\cap{I_n}\subset{D}$.

\end{itemize}
To see it, define two sequences $\la{n_k \colon k<\omega}\ra$ and
$\la{S_{k}\in{\perf{(\Q)}} \colon k<\omega}\ra$
such that $S_k\subset{D_k\cap{I_{n_{k}}}\cap{J_k}}$ where $J_k$ is a clopen
interval
such that $p\notin{\text{cl}_{\real}(J_k)}$.
If $n_k$ and $S_k$ are already defined pick $n_{k+1}>n_k$ with
$J_k\cap{I_{n_{k+1}}}=\emptyset$.
Since $D_{k+1}\cap{I_{n_{k+1}}}\in{F(\D)}$
and $F(\D)$ is crowded we can find a clopen interval $J_{k+1}$ such that
$p\notin{\text{cl}_{\real}(J_{k+1})}$ and
$J_{k+1}\cap{D_{k+1}\cap{I_{n_{k+1}}}}\neq{\emptyset}$.
Define \linebreak
$S_{k+1}=J_{k+1}\cap{D_{k+1}\cap{I_{n_{k+1}}}}$. Then, $S_{k+1}\in{\perf{(\Q)}}$
and $S_{k+1}\subset{D_{k+1}\cap{I_{n_{k+1}}}}$.
Now, put $S=\bigcup_{k<\omega}S_k$. Then, $S\in{\perf{(\Q)}}$ and
$S\cap{I_{n_k}}={\bigcup_{i\geq{k}}{S_i}\cap{I_{n_k}}}=
\bigcup_{i\geq{k}}S_i\subset{D_k}$. This proves our claim.

Let $\B$ be a countable basis for the topology on $\Q$ consisting
of clopen sets and consider the family $\B_0=\{B\in{\B} \colon
|B\cap{S}|=\omega\}$.

If $P\in\perf(\P)$  apply Lemma \ref{lem:lema1} to $P$ and the family
$\{B\cap{S} \colon B\in{\B_0}\}$ to find a set $T\in{[S]^{\omega}}$
and a subprism $Q$ of $P$ such that
\begin{itemize}
\item[(a)]{$|T\cap{(B\cap{S})}|=\omega$ for every $B\in{\B_0}$ and}
\item[(b)]{$|T\cap{x(k)}|\leq{1}$ for every $x\in{Q}$ and $k\in{\omega}$.}
\end{itemize}
If $P=\{x\}$ is a singleton we put $Q=P$ and apply Lemma \ref{lem:lema1}
to the family $\{B\cap{S} \colon B\in{\B_0}\}$ and to $x$ to obtain a $T$
satisfying (a)
and (b).

In both cases we obtain from (a) that $T$ is dense in $S$.
Since $S_k\in{\perf{(\Q)}}$ for every $n<\omega$ we conclude that $T\cap{S_k}$
is non-scattered and contains a subset $Z_k$ from $\perf{(\Q)}$ for every
$k<\omega$.
Hence, if we put $Z=\bigcup_{k<\omega}Z_k$ then, $Z\in{\perf{(\Q)}}$,
$Z\cap{I_k}\subset{D_k}$ for every $k<\omega$ and $|Z\cap{x(k)}|\leq{1}$
for every $x\in{Q}$ and every $k<\omega$. To see that $F(\D\cup{\{Z\}})$
is crowded note that $Z\cap{D_{n_k}}\subset{S\cap{I_{n_k}}}\subset{D_k}$
for every $k<\omega$.
\qed


\thm{thm:teorema3}{\psmPrGame \; implies that there exists an
$\omega_1$-generated crowded $Q$-point on $\Q$.}
%
\proof For $\Y=[\Q]^\omega\cup{\P}$ consider the topology $\tau$ on $\Y$
whose open sets are those $U\subset{\Y}$ such that $U\cap{[\Q]^\omega}$
and $U\cap{\P}$ are open in $[\Q]^\omega$ and $\P$ respectively.
Then $\la{\Y,\tau}\ra$ is a Polish space.
Note that $[\Q]^\omega$ and $\P$ are clopen in $\Y$ with this
topology.
Every prism $P\in{\perf{(\Y)}}$ must intersect either $[\Q]^\omega$
or $\P$. Since every non-empty clopen set in a prism is its subprism
(see \cite{book}, or use Proposition~\ref{prop:teorema1}) we can suppose
without any
loss of generality that either $P\in{\perf([\Q]^\omega)}$ or
$P\in{\perf{(\P)}}$.
Of course, every singleton is in either $[\Q]^\omega$ or $\P$.
Therefore, given a prism $P$ in $\Y$ and a countable family
$\D\subset{\perf{(\Q)}}$
such that $F(\D)$ is crowded we denote by $Z(\D,P)\in{\perf{(\Q)}}$ and
a subprism $Q(\D,P)$ of $P$ as in Lemma \ref{lem:lema4}
provided $P\subset[\Q]^\omega$
and as in Lemma~\ref{lem:lema2} for $P\subset \P$ respectively.

Consider the following strategy $S$ for Player II:
$$S(\la{\la{P_{\eta,},Q_{\eta}}\ra \colon \eta<\xi}\ra,P_{\xi})
=Q(Z(\{Z_{\eta} \colon \eta<\xi\}),P_{\xi}),$$
where sets $Z_{\eta}$ are defined inductively by
$Z_{\eta}=Z(\{Z_{\zeta} \colon \zeta<\eta\},P_{\eta})$.

By \psmPrGame \;  strategy $S$ is not a winning strategy for Player II.
Hence, there is a game $\la{\la{P_{\xi},Q_{\xi}}\ra \colon \xi<\omega_1}\ra$
played according to $S$ for which Player~II loses so,
$\Y=\bigcup_{\xi<\omega_1}Q_{\xi}$.

Let $\U=F(\{Z_{\xi} \colon \xi<\omega_1\})$.
To see it is an ultrafilter note that if $x\in[\Q]^{\omega}$ then there exists
a $\xi<\omega_1$ such that $x\in{Q_{\xi}}$. But then, either
$Z_{\xi}\subset{x}$ or
$Z_{\xi}\cap{x}=\emptyset$. Therefore either $x$ or its complement
is in $\U$. This proves that $\U$ is an ultrafilter and that
$\la{Z_{\xi} \colon \xi<\omega_1}\ra\subset{\perf{(\Q)}}$ is basis
for $\U$. So, $\U$ is crowded. Since, no crowded ultrafilter can be principal
it follows that $\U$ is also non-principal.
To see that $\U$ is a $Q$-point, pick an $x\in{\P}$.
Then, there exists a $\xi<\omega_1$ such that $x\in{Q_{\xi}}$. Thus,
$Z_{\xi}\in{\U}$ and $|Z_{\xi}\cap{x(k)}|\leq{1}$ for every $k<\omega$.
\qed



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\bibitem{book}
Ciesielski, K. and Pawlikowski, J.
\emph{Covering Property Axiom \cpa}, to appear in Cambridge Tracts in
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\bibitem{crowded2}
Coplakova, E. and Hart, K.P.

\emph{Crowded rational ultrafilters},
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\bibitem{crowded1}
van Douwen, E.K.
\emph{Better closed ultrafilters on $\Q$},
Topology Appl. \textbf{47} 1992, 173-177.

\bibitem{SetTheory}

Jech, T.
\emph{Set Theory},
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\bibitem{Borel}
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Acta Math. \textbf{137}, 1976, 151-169.

\bibitem{qpoint}
Miller, A.W.
\emph{There are no $Q$-points in Laver's model for the Borel
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\end{thebibliography}

\end{document}