From f35ae521b3542cc7533c633f3c00c591a0d71fe7 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Wed, 9 Feb 2022 20:04:34 +0100
Subject: pierwsza strona, pora na spacer

---
 lic_malinka.tex | 54 ++++++++++++++++++++++++++++++++++++++++++------------
 1 file changed, 42 insertions(+), 12 deletions(-)

(limited to 'lic_malinka.tex')

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
-- 
cgit v1.2.3