Aspelled

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.