From e55ffead297fd04fe73e5f7bd6d05a151450fb99 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sat, 9 Jul 2022 13:44:59 +0200
Subject: Aspelled

---
 sections/preliminaries.tex | 45 +++++++++++++++++++++++++++++++++------------
 1 file changed, 33 insertions(+), 12 deletions(-)

(limited to 'sections/preliminaries.tex')

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.
 
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