diff options
Diffstat (limited to 'sections')
| -rw-r--r-- | sections/conj_classes.tex | 5 | ||||
| -rw-r--r-- | sections/fraisse_classes.tex | 9 | ||||
| -rw-r--r-- | sections/preliminaries.tex | 45 |
3 files changed, 41 insertions, 18 deletions
diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex index 8beace0..3e4cb3a 100644 --- a/sections/conj_classes.tex +++ b/sections/conj_classes.tex @@ -91,7 +91,7 @@ that $\Pi = \Gamma$. We will construct a strategy for the second player in the Banach-Mazur game on the topological space $G$. This strategy will give us a subset - $A\subseteq G$ and as we will see a subset of a cojugacy class in $G$. + $A\subseteq G$ and as we will see a subset of a conjugacy class in $G$. By the Banach-Mazur theorem \ref{theorem:banach_mazur_thm} this will prove that this class is comeagre. @@ -234,6 +234,7 @@ % infinite and has the random graph property. % \end{proof} \begin{proposition} + \label{proposition:fixed_points} Let $\sigma$ be the generic automorphism of $\Gamma$. Then the set of fixed points of $\sigma$ is isomorphic to $\Gamma$. \end{proposition} @@ -242,7 +243,7 @@ Let $S = \{x\in \Gamma\mid \sigma(x) = x\}$. First we need to show that it is an infinite. By the theorem \ref{theorem:generic_aut_general} we know that $(\Gamma, \sigma)$ is the Fraïssé limit of $\cH$, thus we - can embedd finite $L$-structures of any size with identity as an + can embed finite $L$-structures of any size with identity as an automorphism of the structure into $(\Gamma, \sigma)$. Thus $S$ has to be infinite. Also, the same argument shows that the age of the structure is exactly $\cC$. It is weakly ultrahomogeneous, also by the fact that diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex index 9110f44..dc6392d 100644 --- a/sections/fraisse_classes.tex +++ b/sections/fraisse_classes.tex @@ -149,6 +149,7 @@ \end{proof} \begin{definition} + \label{definition:random_graph} The \emph{random graph} is the Fraïssé limit of the class of finite graphs $\cG$ denoted by $\FrGr = \Flim(\cG)$. \end{definition} @@ -253,7 +254,7 @@ \end{tikzcd} \end{center} - We have deliberately omited names for embeddings of $C$. Of course, + We have deliberately omitted names for embeddings of $C$. Of course, the functor has to take them into account, but this abuse of notation is convenient and should not lead into confusion. \item Let $A\leftarrow C\rightarrow B$, $A'\leftarrow C\rightarrow B'$ be cospans @@ -414,13 +415,13 @@ theorem \ref{theorem:fraisse_thm} it suffices to show that the age of $\Pi$ is $\cC$ and that it has the weak ultrahomogeneity in the class $\cC$. The former comes easily, as for every structure $A\in \cC$ we have the structure - $(A, \id_A)\in \cD$, which means that the structure $A$ embedds into $\Pi$. - Also, if a structure $(B, \beta)\in\cD$ embedds into $\cD$, then $B\in\cC$. + $(A, \id_A)\in \cD$, which means that the structure $A$ embeds into $\Pi$. + Also, if a structure $(B, \beta)\in\cD$ embeds into $\cD$, then $B\in\cC$. Hence, $\cC$ is indeed the age of $\Pi$. Now, take any structure $A, B\in\cC$ such that $A\subseteq B$. Without the loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}$ be the - smallest structure closed on the automorphism $\sigma$ and containg $A$. + smallest structure closed on the automorphism $\sigma$ and containing $A$. It is finite, as $\cC$ is the age of $\Pi$. By the weak Hrushovski property, of $\cC$ let $(\bar{B}, \beta)$ be a structure extending $(B\cup \bar{A}, \sigma\upharpoonright_{\bar{A}})$. Again, we may assume diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex index 832beae..5f53c0a 100644 --- a/sections/preliminaries.tex +++ b/sections/preliminaries.tex @@ -1,7 +1,27 @@ \documentclass[../lic_malinka.tex]{subfiles} \begin{document} + Before we get to the main work of the paper, we need to establish basic + notions, known facts and theorems. This section provides a brief + introduction to the theory of Baire spaces and category theory. + Most of the notions are well known, interested reader may look at + \cite{descriptive_set_theory}, \cite{maclane_1978} + \subsection{Descriptive set theory} + In this section we provide an important definition of a \emph{comeagre} set. + It is purely topological notion, the intuition may come from the measure + theory though. For example, in a standard Lebesuge measure on the + real interval $[0,1]$, the set of rationals is of measure $0$, although + being a dense subset of the $[0,1]$. So, in a sense, the set of rationals + is \emph{meagre} in the interval $[0,1]$. On the other hand, the set + of irrational numbers is also dense, but have measure $1$, so it is + \emph{comeagre}. + + This is only a rough approximation of the topological + definition. The definitions are based on the Kechris' book \textit{Classical + Descriptive Set Theory} \cite{descriptive_set_theory}. One should look into + it for more details and examples. + \begin{definition} Suppose $X$ is a topological space and $A\subseteq X$. We say that $A$ is \emph{meagre} in $X$ if $A = \bigcup_{n\in\bN}A_n$, @@ -15,11 +35,10 @@ 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, $\bQ$ + Every countable set is meagre in any $T_1$ space. So, $\bQ$ is meagre in $\bR$ (although it is dense), which means that the set of - irrationals is comeagre. Another example is... - % \end{example} + irrationals is comeagre. The Cantor set is nowhere dense, hence meagre + in the $[0,1]$ interval. \begin{definition} We say that a topological space $X$ is a \emph{Baire space} if every @@ -43,10 +62,10 @@ $\bigcap_{n}V_n \subseteq A$. \end{definition} - There is an important theorem on the Banach-Mazur game: $A$ is comeagre - if and only if $\textit{II}$ 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. + There is an important theorem \ref{theorem:banach_mazur_thm} on the + Banach-Mazur game: $A$ is comeagre if and only if $\textit{II}$ 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 @@ -96,16 +115,18 @@ $\bigcap_{n}V_n \subseteq A$. \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} + + The statement of the theorem is once again taken from Kechris + \cite{descriptive_set_theory} 8.33. However, the proof given in the book is + brief, thus we present a detailed version. In order to prove the + theorem we add an auxiliary definition and lemma. - In order to prove it we add an auxiliary 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 @@ -336,7 +357,7 @@ \end{definition} \begin{definition} - The \emph{cospan category} of category $\cC$, refered to as $\Cospan(\cC)$, + The \emph{cospan category} of category $\cC$, referred to as $\Cospan(\cC)$, is the category of cospan diagrams of $\cC$, where morphisms between two cospans are normal transformations of the underlying functors. |