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\title{Tytuł} 
\author{Franciszek Malinka}

\begin{document}
  \begin{abstract} 
    Abstract 
  \end{abstract} 
  \section{Introduction}

  \section{Preliminaries} 
  \subsection{Descriptive set theory}
  \begin{definition} 
    Suppose $X$ is a topological space and $A\subseteq X$.
    We say that $A$ is \emph{meagre} in $X$ if $A = \bigcup_{n\in\bN}A_n$,
    where $A_n$ are nowhere dense subsets of $X$ (i.e. $\Int(\bar{A_n})
    = \emptyset$).  
  \end{definition}

  \begin{definition} 
    We say that $A$ is \emph{comeagre} in $X$ if it is
    a complement of a meagre set. Equivalently, a set is comeagre iff it
    contains a countable intersection of open dense sets.  
  \end{definition}

      % \begin{example}
  Every countable set is meagre in any $T_1$ space, so, for example, $\bQ$
  is meagre in $\bR$ (though being dense), which means that the set of
  irrationals is comeagre. Another example is...
      % \end{example}

  \begin{definition}
    We say that a topological space $X$ is a \emph{Baire space} if every
    comeagre subset of $X$ is dense in $X$ (equivalently, every meagre set has
    empty interior).  
  \end{definition}

  \begin{definition}
    Suppose $X$ is a Baire space. We say that a property $P$ \emph{holds
    generically} for a point in $x\in X$ if $\{x\in X\mid P\textrm{ holds for
    }x\}$ is comeagre in $X$.  
  \end{definition}

  \begin{definition} Let $X$ be a nonempty topological space and let
    $A\subseteq X$. The \emph{Banach-Mazur game of $A$}, denoted as
    $G^{\star\star}(A)$ is defined as follows: Players $I$ and
    $\textit{II}$ take turns in playing nonempty open sets $U_0, V_0,
    U_1, V_1,\ldots$ such that $U_0 \supseteq V_0 \supseteq U_1 \supseteq
    V_1 \supseteq\ldots$. We say that player $\textit{II}$ wins the game if
    $\bigcap_{n}V_n \subseteq A$.  
  \end{definition}

  There is an important theorem on the Banach-Mazur game: $A$ is comeagre
  iff $\textit{II}$ can always choose sets $V_0, V_1, \ldots$ such that
  it wins. Before we prove it we need to define notions necessary to
  formalise and prove the theorem.

  \begin{definition}
    $T$ is \emph{the tree of all legal positions} in the Banach-Mazur game
    $G^{\star\star}(A)$ when $T$ consists of all finite sequences $(W_0,
    W_1,\ldots, W_n)$, where $W_i$ are nonempty open sets such that
    $W_0\supseteq W_1\supseteq\ldots\supseteq W_n$. In another words, $T$ is
    a pruned tree on $\{W\subseteq X\mid W \textrm{is open nonempty}\}$.
  \end{definition} 

  \begin{definition}
    We say that $\sigma$ is \emph{a pruned subtree} of the tree of all legal
    positions $T$ if $\sigma\subseteq T$ and for any $(W_0, W_1, \ldots,
    W_n)\in\sigma, n\ge 0$ there is a $W$ such that $(W_0, W_1,\ldots, W_n,
    W)\in\sigma$ (it simply means that there's no finite branch in~$\sigma$).  
  \end{definition}

  \begin{definition}
    Let $\sigma$ be a pruned subtree of the tree of all legal positions $T$. By
    $[\sigma]$ we denote \emph{the set of all infinite branches of $\sigma$},
    i.e. infinite sequences $(W_0, W_1, \ldots)$ such that $(W_0, W_1, \ldots
    W_n)\in \sigma$ for any $n\in \bN$.  
  \end{definition}

  \begin{definition} 
    A \emph{strategy} for $\textit{II}$ in
    $G^{\star\star}(A)$ is a pruned subtree $\sigma\subseteq T$ such that
    \begin{enumerate}[label=(\roman*)] 
      \item $\sigma$ is nonempty, 
      \item if $(U_0, V_0, \ldots, U_n, V_n)\in\sigma$, then for all open
      nonempty $U_{n+1}\subseteq V_n$, $(U_0, V_0, \ldots, U_n, V_n,
      U_{n+1})\in\sigma$, 
      \item if $(U_0, V_0, \ldots, U_{n})\in\sigma$, then for a unique $V_n$,
        $(U_0, V_0, \ldots,  U_{n}, V_n)\in\sigma$.
    \end{enumerate} 
  \end{definition}

  Intuitively, a strategy $\sigma$ works as follows: $I$ starts playing
  $U_0$ as any open subset of $X$, then $\textit{II}$ plays unique (by
  (iii)) $V_0$ such that $(U_0, V_0)\in\sigma$. Then $I$ responds by
  playing any $U_1\subseteq V_0$ and $\textit{II}$ plays uniqe $V_1$
  such that $(U_0, V_0, U_1, V_1)\in\sigma$, etc.

  \begin{definition} 
    A strategy $\sigma$ is a \emph{winning strategy for $\textit{II}$} if for
    any game $(U_0, V_0\ldots)\in [\sigma]$ player $\textit{II}$ wins, i.e.
    $\bigcap_{n}V_n \subseteq A$.
  \end{definition}

  Now we can state the key theorem.

  \begin{theorem}[Banach-Mazur, Oxtoby]
  \label{theorem:banach_mazur_thm} 
    Let $X$ be a nonempty topological space and let $A\subseteq X$. Then A is
    comeagre $\Leftrightarrow$ $\textit{II}$ has a winning strategy in
    $G^{\star\star}(A)$.
  \end{theorem}

  In order to prove it we add an auxilary definition and lemma.
  \begin{definition}
    Let $S\subseteq\sigma$ be a pruned subtree of tree of all legal positions
    $T$ and let $p=(U_0, V_0,\ldots, V_n)\in S$.  We say that S is
    \emph{comprehensive for p} if the family $\cV_p = \{V_{n+1}\mid (U_0,
    V_0,\ldots, V_n, U_{n+1}, V_{n+1})\in S\}$ (it may be that $n=-1$, which
    means $p=\emptyset$) is pairwise disjoint and $\bigcup\cV_p$ is dense in
    $V_n$ (where we think that $V_{-1} = X$).
    We say that $S$ is \emph{comprehensive} if it is comprehensive for
    each $p=(U_0, V_0,\ldots, V_n)\in S$.  
  \end{definition}

  \begin{fact} 
    If $\sigma$ is a winnig strategy for $\mathit{II}$ then
    there exists a nonempty comprehensive $S\subseteq\sigma$. 
  \end{fact}

  \begin{proof}
    We construct $S$ recursively as follows:
    \begin{enumerate} 
      \item $\emptyset\in S$, 
      \item if $(U_0, V_0, \ldots, U_n)\in S$, then $(U_0, V_0, \ldots, U_n,
        V_n)\in S$ for the unique $V_n$ given by the strategy $\sigma$, 
      \item let $p = (U_0, V_0, \ldots, V_n)\in S$. For a possible player $I$'s
        move $U_{n+1}\subseteq V_n$ let $U^\star_{n+1}$ be the unique set
        player $\mathit{II}$ would respond with by $\sigma$. Now, by Zorn's
        Lemma, let $\cU_p$ be a maximal collection of nonempty open subsets
        $U_{n+1}\subseteq V_n$ such that the set $\{U^\star_{n+1}\mid
        U_{n+1}\in\cU_p\}$ is pairwise disjoint.  Then put in $S$ all $(U_0,
        V_0, \ldots, V_{n}, U_{n+1})$ such that $U_{n+1} \in \cU_p$. This way
        $S$ is comprehensive for $p$: the family $\cV_p = \{V_{n+1}\mid (U_0,
        V_0,\ldots, V_n, U_{n+1}, V_{n+1})\in S\}$ is exactly
        $\{U^\star_{n+1}\mid U_{n+1}\in\cU_p\}$, which is pairwise disjoint and
        $\bigcup\cV_p$ is obviously dense in $V_n$ by the maximality of $\cU_p$
        -- if there was any open set $\tilde{U}_{n+1}\subseteq V_n$ disjoint
        from $\bigcup\cV_p$, then $\tilde{U}^{\star}_{n+1}\subseteq
        \tilde{U}_{n+1}$ would be also disjoint from $\bigcup\cV_p$, so the
        family $\cU_p\cup\{\tilde{U}_{n+1}\}$ would violate the maximality of
        $\cU_p$.  
      \qedhere 
    \end{enumerate} 
  \end{proof}

  \begin{lemma} 
    \label{lemma:comprehensive_lemma}
    Let $S$ be a nonempty comprehensive pruned subtree of a strategy $\sigma$.
    Then:
    \begin{enumerate}[label=(\roman*)]
      \item For any open $V_n\subseteq X$ there is at most one $p=(U_0, V_0,
        \ldots, U_n, V_n)\in S$.  
      \item Let $S_n = \{V_n\mid (U_0, V_0, \ldots, V_n)\in S\}$ for $n\in\bN$
        (i.e. $S_n$ is a family of all possible choices player $\textit{II}$
        can make in its $n$-th move according to $S$). Then $\bigcup S_n$ is
        open and dense in $X$.  
      \item $S_n$ is a family of pairwise disjoint sets.  
    \end{enumerate} 
  \end{lemma}

  \begin{proof} 
    (i): Suppose that there are some $p = (U_0, V_0,\ldots,
    U_n, V_n)$, $p'=(U'_0, V'_0, \ldots, U'_n, V'_n)$ such that $V_n
    = V'_n$ and $p \neq p'$. Let $k$ be the smallest index such that those
    sequences differ. We have two possibilities: 
    \begin{itemize} 
      \item $U_k = U'_k$ and $V_k\neq V'_k$ -- this cannot be true simply by
        the fact that $S$ is a subset of a strategy (so $V_k$ is unique for
        $U_k$). \item $U_k\neq U'_k$: by the comprehensiveness of $S$ we know 
        that for $q =(U_0, V_0, \ldots, U_{k-1}, V_{k-1})$ the set $\cV_q$ is 
        pairwise disjoint. Thus $V_k\cap V'_k=\emptyset$, because $V_k, V'_k\in 
        \cV_q$. But this leads to a contradiction -- $V_n$ cannot be a nonempty 
        subset of both $V_k, V'_k$.  
    \end{itemize}

    (ii): The lemma is proved by induction on $n$. For $n=0$ it follows
    trivially from the definition of comprehensiveness. Now suppose the
    lemma is true for $n$. Then the set $\bigcup_{V_n\in
    S_n}\bigcup\cV_{p_{V_n}}$ (where $p_{V_n}$ is given uniquely from
    (i)) is dense and open in $X$ by the induction hypothesis. But
    $\bigcup S_{n+1}$ is exactly this set, thus it is dense and open in
    $X$.

    (iii): We will prove it by induction on $n$. Once again, the case $n
    = 0$ follows from the comprehensiveness of $S$. Now suppose that the
    sets in $S_n$ are pairwise disjoint. Take some $x \in V_{n+1}\in
    S_{n+1}$. Of course $\bigcup S_n \supseteq \bigcup S_{n+1}$, thus by
    the inductive hypothesis $x\in V_{n}$ for the unique $V_n\in S_n$. It
    must be that $V_{n+1}\in \cV_{p_{V_n}}$, because $V_n$ is the only
    superset of $V_{n+1}$ in $S_n$. But $\cV_{p_{V_n}}$ is disjoint, so
    there is no other $V'_{n+1}\in \cV_{p_{V_n}}$ suc h that $x\in
    V'_{n+1}$. Moreover, there is no such set in
    $S_{n+1}\setminus\cV_{p_{V_n}}$, because those sets are disjoint from
    $V_{n}$. Hence there is no $V'_{n+1}\in S_{n+1}$ other than $V_n$
    such that $x\in V'_{n+1}$. We've chosen $x$ and $V_{n+1}$ arbitrarily,
    so $S_{n+1}$ is pairwise disjoint.  
  \end{proof}

  Now we can move to the proof of the Banach-Mazur theorem.

  \begin{proof}[Proof of theorem \ref{theorem:banach_mazur_thm}]
    $\Rightarrow$: Let $(A_n)$ be a sequence of dense open sets with
    $\bigcap_n A_n\subseteq A$.  The simply $\textit{II}$ plays $V_n
    = U_n\cap A_n$, which is nonempty by the denseness of $A_n$.

    $\Leftarrow$: Suppose $\textit{II}$ has a winning strategy $\sigma$.
    We will show that $A$ is comeagre. Take a comprehensive $S\subseteq
    \sigma$. We claim that $\mathcal{S} = \bigcap_n\bigcup S_n \subseteq
    A$. By the lemma~\ref{lemma:comprehensive_lemma}, (ii) sets $\bigcup
    S_n$ are open and dense, thus $A$ must be comeagre. Now we prove the
    claim towards contradiction. 
  
    Suppose there is $x\in \mathcal{S}\setminus A$. By the lemma
    \ref{lemma:comprehensive_lemma}, (iii) for any $n$ there is unique
    $x\in V_n\in S_n$. It follows that $p_{V_0}\subset
    p_{V_1}\subset\ldots$. Now the game $(U_0, V_0, U_1, V_1,\ldots)
    = \bigcup_n p_{V_n}\in [S]\subseteq [\sigma]$ is not winning for
  player $\textit{II}$, which contradicts the assumption that $\sigma$ is
  a winning strategy.  \end{proof}

  \begin{corollary} 
    \label{corollary:banach-mazur-basis} 
    If we add a constraint to the Banach-Mazur game such that players can only 
    choose basic open sets, then the theorem \ref{theorem:banach_mazur_thm} 
    still suffices.  
  \end{corollary}

  \begin{proof} 
    If one adds the word \textit{basic} before each occurrence
    of word \textit{open} in previous proofs and theorems then they all
    will still be valid (except for $\Rightarrow$, but its an easy fix --
    take $V_n$ a basic open subset of $U_n\cap A_n$).  
  \end{proof}

  This corollary will be important in using the theorem in practice --
  it's much easier to work with basic open sets rather than any open
  sets.

  \section{Fraïssé classes}

  In this section we will take a closer look at classes of finitely
  generated structures with some characteristic properties. More
  specifically, we will describe a concept developed by a French
  mathematician Roland Fraïssé called Fraïssé limit.

  \subsection{Definitions} 
  \begin{definition}
    Let $L$ be a signature and $M$ be an $L$-structure. The \emph{age} of $M$ is
    the class $\bK$ of all finitely generated structures that embedds into $M$.
    The age of $M$ is also associated with class of all structures embeddable in
    $M$ \emph{up to isomorphism}.
  \end{definition}

  \begin{definition}
    We say that $M$ has \emph{countable age} when its age has countably many 
    isomorphism types of finitely generated structures. 
  \end{definition}

  \begin{definition}
    Let $\bK$ be a class of finitely generated structures. $\bK$ has 
    \emph{hereditary property (HP)} if for any $A\in\bK$, any finitely generated
    substructure $B$ of $A$ it holds that $B\in\bK$. 
  \end{definition}

  \begin{definition}
    Let $\bK$ be a class of finitely generated structures. We say that $\bK$ has
    \emph{joint embedding property (JEP)} if for any $A, B\in\bK$ there is a
    structure $C\in\bK$ such that both $A$ and $B$ embed in $C$.
  \end{definition}

  Fraïssé has shown fundamental theories regarding age of a structure, one of 
  them being the following one:
  
  \begin{fact}
    \label{fact:age_is_hpjep}
    Suppose $L$ is a signature and $\bK$ is a nonempty finite or countable set 
    of finitely generated $L$-structures. Then $\bK$ has the HP and JEP if
    and only if $\bK$ is the age of some finite or countable structure. 
  \end{fact}

  Beside the HP and JEP Fraïssé has distinguished one more property of the 
  class $\bK$, namely amalgamation property.

  \begin{definition}
    Let $\bK$ be a class of finitely generated $L$-structures. We say that $\bK$
    has the \emph{amalgamation property (AP)} if for any $A, B, C\in\bK$ and
    embeddings $f\colon A\to B, g\colon A\to C$ there exists $D\in\bK$ together
    with embeddings $h\colon B\to D$ and $j\colon C\to D$ such that 
    $h\circ f = j\circ g$.
    \begin{center}
      \begin{tikzcd}
                                  & D & \\
        B \arrow[ur, dashed, "h"] &   & C \arrow[ul, dashed, "j"'] \\
                                  & A \arrow[ur, "g"'] \arrow[ul, "f"]
      \end{tikzcd}
    \end{center}
  \end{definition}

  \begin{definition}
    Let $M$ be an $L$-structure. $M$ is \emph{ultrahomogenous} if every
    isomorphism between finitely generated substrucutres of $M$ extends to an
    automorphism of $M$.
  \end{definition}

  Having those definitions we can provide the main Fraïssé theorem.

  \begin{theorem}[Fraïssé theorem]
    \label{theorem:fraisse_thm}
    Let L be a countable language and let $\bK$ be a nonempty countable set of
    finitely generated $L$-structures which has HP, JEP and AP. Then $\bK$ is 
    the age of a countable, ultrahomogenous $L$-structure $M$. Moreover, $M$ is
    unique up to isomorphism. We say that $M$ is a \emph{Fraïssé limit} of $\bK$
    and denote this by $M = \Flim(\bK)$.
  \end{theorem}

  This is a well known theorem. One can read a proof of this theorem in Wilfrid
  Hodges' classical book \textit{Model Theory}~\cite{hodges_1993}. In the proof
  of this theorem appears another, equally important \ref{lemma:weak_ultrahom}.

  \begin{definition}
    We say that an $L$-structure $M$ is \emph{weakly ultrahomogenous} if for any
    $A, B$ finitely generated substructures of $M$ such that $A\subseteq B$ and
    an embedding $f\colon A\to M$ there is an embedding $g\colon B\to M$ which
    extends $f$.
    \begin{center}
      \begin{tikzcd}
        A \arrow[d, "\subseteq"'] \arrow[r, "f"] & D \\
        B \arrow[ur, dashed, "g"']
      \end{tikzcd}
    \end{center}
  \end{definition}

  \begin{lemma}
    \label{lemma:weak_ultrahom}
    A countable structure is ultrahomogenous if and only if it is weakly
    ultrahomogenous.
  \end{lemma}

  This lemma will play a major role in the later parts of the paper. Weak
  ultrahomogenity is an easier and more intuitive property and it will prove
  useful when recursively constructing the generic automorphism of a Fraïssé 
  limit.


  % \begin{definition} For $\cC$, $M$ as in Fact~\ref{fact:fraisse_thm}, we
  % write $\Flim(\cC)\coloneqq M$.  \end{definition}

  % \begin{fact} If $\cC$ is a uniformly locally finite Fraïssé class, then
  % $\Flim(\cC)$ is $\aleph_0$-categorical and has quantifier elimination.
  % \end{fact}

  % \section{Conjugacy classes in automorphism groups}

  % \subsection{Prototype: pure set} In this section, $M=(M,=)$ is an
  % infinite countable set (with no structure beyond equality).
  % \begin{proposition} If $f_1,f_2\in \Aut(M)$, then $f_1$ and $f_2$ are
  %   conjugate if and only if for each $n\in \bN\cup \{\aleph_0\}$, $f_1$
  %   and $f_2$ have the same number of orbits of size $n$.
  % \end{proposition}

  % \begin{proposition} The conjugacy class of $f\in \Aut(M)$ is dense if
  %   and only if...  \end{proposition} \begin{proposition} If $f\in
  % \Aut(M)$ has an infinite orbit, then the conjugacy class of $f$ is
  % meagre.  
  % \end{proposition}

  % \begin{proposition} An automorphism $f$ of $M$ is generic if and only if...
  % \end{proposition}

  % \begin{proof}

  % \end{proof}

  % \subsection{More general structures}


  % \begin{proposition} Suppose $M$ is an arbitrary structure and $f_1,f_2\in
  % \Aut(M)$. Then $f_1$ and $f_2$ are conjugate if and only if $(M,f_1)\cong
  % (M,f_2)$.  \end{proposition}

  % \begin{definition} We say that a Fraïssé class $\cC$ has \emph{weak
  %   Hrushovski property} (\emph{WHP}) if for every $A\in \cC$ and partial
  %   automorphism $p\colon A\to A$, there is some $B\in \cC$ such that $p$ can
  %   be extended to an automorphism of $B$, i.e.\ there is an embedding
  %   $i\colon A\to B$ and a $\bar p\in \Aut(B)$ such that the following
  %   diagram commutes: 
  %   \begin{center} 
  %     \begin{tikzcd} 
  %       B\ar[r,"\bar p"]&B\\
  %       A\ar[u,"i"]\ar[r,"p"]&A\ar[u,"i"] 
  %     \end{tikzcd} 
  %   \end{center}
  % \end{definition}

  % \begin{proposition} Suppose $\cC$ is a Fraïssé class in a relational
  % language with WHP. Then generically, for an $f\in \Aut(\Flim(\cC))$, all
  % orbits of $f$ are finite.  \end{proposition} \begin{proposition} Suppose
  %   $\cC$ is a Fraïssé class in an arbitrary countable language with WHP.
  %   Then generically, for an $f\in \Aut(\Flim(\cC))$ ...  \end{proposition}

  % \subsection{Random graph} \begin{definition} The \emph{random graph}
  % is...  \end{definition}

  % \begin{fact} The \end{fact}

  %   \begin{proposition} Generically, the set of fixed points of $f\in
  %   \Aut(M)$ is isomorphic to $M$ (as a graph).  \end{proposition}
  \printbibliography
\end{document}