pierwsza strona, pora na spacer

1 files changed, 42 insertions, 12 deletions

diff --git a/lic_malinka.tex b/lic_malinka.tex
index 7abd37d..adaaf91 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -6,7 +6,7 @@
 \usepackage{mathtools}

 \usepackage[activate={true,nocompatibility},final,tracking=true,kerning=true,spacing=true,stretch=10,shrink=10]{microtype}

 \microtypecontext{spacing=nonfrench}

-\usepackage[utf8]{inputenc}

+% \usepackage[utf8]{inputenc}

 

 \usepackage{amsfonts}

 \usepackage{amsmath}

@@ -52,14 +52,15 @@
 

 \newcommand{\cupdot}{\mathbin{\mathaccent\cdot\cup}}

 \newcommand{\cC}{\mathcal C}

-\newcommand{\bN}{\mathbf N}

-\newcommand{\bR}{\mathbf R}

-\newcommand{\bZ}{\mathbf Z}

+\newcommand{\bN}{\mathbb N}

+\newcommand{\bR}{\mathbb R}

+\newcommand{\bZ}{\mathbb Z}

+\newcommand{\bQ}{\mathbb Q}

 

 \DeclareMathOperator{\im}{{Im}}

 \DeclareMathOperator{\lin}{{Lin}}

 \DeclareMathOperator{\Th}{{Th}}

-

+\DeclareMathOperator{\Int}{{Int}}

 

 \newtheorem{theorem}{Theorem}

 \numberwithin{theorem}{section}

@@ -113,25 +114,54 @@
     \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... Wea say that $A$ is \emph{comeagre} in $X$ if... .

+        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 meager set. Equivalently, a set is comeagre iff it contains a countable intersection of open dense sets.

+    \end{definition}

+    

+    % \begin{example}

+    Every countable set is nowhere dense in any $T_1$ space, so, for example, $\bQ$ is meager 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 $P$ holds generically for a point in $x\in X$ if $\{x\in X\mid P\textrm{ holds for }x\}$ is comeagre in $X$.

+        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 $II$ 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 $II$ 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 $II$ can always choose sets $V_0, V_1, \ldots$ such that it wins. Before we prove it we need to define notions necessary to formalize this 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 $\{W\subseteq X\mid W \textrm{is open nonempty}\}$. 

+    

+        By $[T]$ we denote the set of all "infinite branches" of $T$, i.e. infinite sequences $(U_0, V_0, \ldots)$ such that $(U_0, V_0, \ldots U_n, V_n)\in T$ for any $n\in \bN$.

+    \end{definition}

+

+

+    \begin{definition}

+        A \emph{strategy} for $II$ in $G^{\star\star}(A)$ is a subtree $\sigma\subseteq T$ such that

+        \begin{enumerate}[label=(\roman*)]

+            \item $\sigma$ is nonempty,

+            \item if $(U_0, V_0, \ldots, V_n)\in\sigma$, then for all open nonempty $U_{n+1}\subseteq V_n$, $(U_0, V_0, \ldots, V_n, U_{n+1})\in\sigma$,

+            \item if $(U_0, V_0, \ldots, U_{n})\in\sigma$, then for unique $V_n$, $(U_0, V_0, \ldots,  U_{n}, V_n)\in\sigma$. 

+        \end{enumerate}

     \end{definition}

 

-    \begin{example}

-        content

-    \end{example}

+    Intuitively, the strategy $\sigma$ works as follows: $I$ starts playing $U_0$ as any open subset of $X$, then $II$ 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 $II$ plays uniqe $V_1$ such that $(U_0, V_0, U_1, V_1)\in\sigma$, etc.

 

     \subsection{Fraïssé classes}

     \begin{fact}[Fraïssé theorem]

         \label{fact:fraisse_thm}

-        Suppose $\cC$ is a class of finitely generated $L$-structures such that...

+        % Suppose $\cC$ is a class of finitely generated $L$-structures such that...

 

         Then there exists a unique up to isomorphism counable $L$-structure $M$ such that...

     \end{fact}

@@ -205,4 +235,4 @@
         Generically, the set of fixed points of $f\in \Aut(M)$ is isomorphic to $M$ (as a graph).

     \end{proposition}

 

-\end{document}

+\end{document}
\ No newline at end of file