From f35ae521b3542cc7533c633f3c00c591a0d71fe7 Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Wed, 9 Feb 2022 20:04:34 +0100 Subject: pierwsza strona, pora na spacer --- lic_malinka.pdf | Bin 234733 -> 302220 bytes lic_malinka.tex | 54 ++++++++++++++++++++++++++++++++++++++++++------------ 2 files changed, 42 insertions(+), 12 deletions(-) diff --git a/lic_malinka.pdf b/lic_malinka.pdf index 71a04cb..286d1e2 100644 Binary files a/lic_malinka.pdf and b/lic_malinka.pdf differ 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