From 006c57fdeb81d97b1ee222d14346e3844df343f5 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Wed, 13 Jul 2022 23:06:05 +0200
Subject: Chyba poprawiony dowod 4.4

---
 sections/conj_classes.tex | 122 +++++++++++++++++++++++++++++++---------------
 1 file changed, 82 insertions(+), 40 deletions(-)

(limited to 'sections/conj_classes.tex')

diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index 96522f5..9620220 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -82,7 +82,7 @@
 
   \begin{theorem}
 	\label{theorem:generic_aut_general}
-	Let $\cC$ be a Fraïssé class of finite $L$-structures. 
+	Let $\cC$ be a Fraïssé class of finitely generated $L$-structures. 
 	Let $\cD$ be the class of structures from $\cC$ with additional unary 
 	function symbol interpreted
 	as an automorphism of the structure. If $\cC$ has the weak Hrushovski property
@@ -90,11 +90,25 @@
 	automorphism group of the $\Flim(\cC)$.
   \end{theorem}
 
+  Before we get to the proof, let us establish some notions. If 
+  $g\colon\Gamma\to\Gamma$ is a finite injective function, then we say that
+  $g$ is \emph{good} if it gives (in a natural way) an isomorphism between
+  $\langle \dom(g)\rangle$ and $\langle\rng(g)\rangle$, i.e. substructures 
+  generated by $\dom(g)$ and $\rng(g)$ respectively. Of course, in our
+  case, $g$ is good
+  if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (becuase of ultrahomogeneity
+  of $\Gamma$.
+
+  Also it is important to mention that an isomorphism between two finitely
+  generated structures is uniquely given by a map from generators of one structure
+  to the other. This allow us to treat a finite function as an isomorphism
+  of finitely generated structure.
+
   \begin{proof}
-	Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. Let $G = \Aut(\Gamma)$,
-	i.e. $G$ is the automorphism group of $\Gamma$. First, by the Theorem
+	Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. First, by the Theorem
 	\ref{theorem:isomorphic_fr_lims}, we may assume without the loss of generality
-	that $\Pi = \Gamma$.
+	that $\Pi = \Gamma$. Let $G = \Aut(\Gamma)$,
+	i.e. $G$ is the automorphism group of $\Gamma$.
 	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 the $\sigma$'s conjugacy class.
@@ -105,8 +119,8 @@
 	sets $\{g\in G\mid g\upharpoonright_A = g_0\upharpoonright_A\}$ for some
 	finite set $A\subseteq \Gamma$ and some automorphism $g_0\in G$. In other
 	words, a basic open set is a set of all extensions of some finite partial
-	isomorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic 
-	open subset given by a finite partial isomorphism $g$. Note that $B_g$
+	automorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic 
+	open subset given by a finite partial isomorphism $g$. Again, Note that $B_g$
 	is nonemty because of ultrahomogeneity of $\Gamma$.
 
 	With the use of Corollary \ref{corollary:banach-mazur-basis} we can consider 
@@ -150,20 +164,32 @@
 	For technical reasons, let $g_{-1} = \emptyset$ and
 	$X_{-1} = \emptyset$. 
 	Suppose that player \textit{I} in the $n$-th move chooses a finite partial
-	isomorphism $f_n$. We will construct a finite partial isomorphism $g_n\supseteq f_n$ 
-	and a set $X_n\subseteq\bN^2$
-	such that following properties hold:
+	automorphism $f_n$. We will construct a finite partial automorphism 
+	$g_n\supseteq f_n$ together with a finitely generated substrucutre
+	$\Gamma_n \subseteq \Gamma$ and a set $X_n\subseteq\bN^2$ 
+	such that the following properties hold:
 
 	\begin{enumerate}[label=(\roman*)]
-	  \item $g_n$ is an automorphism of the induced substructure $\Gamma_n$,
+	  \item $g_n$ is good and 
+		$\dom(g_n)\cup\rng(g_n)\subseteq\langle\dom(g_n)\rangle = \Gamma_n$,
+		i.e. $g_n$ gives an automorphism of a finitely generated 
+		substructure $\Gamma_n$
 	  \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined,
-	  \item let 
-		$\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$
-		be the enumeration of all pairs of finite structures of $T$ with automorphism 
-		such that the first is a substructure of the second, i.e. 
-		$(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, and $f_{n,k}$
-		is an embedding	of $(A_{n,k}, \alpha_{n,k})$ in the $\FrGr_{n-1}$ (which
-		is the substructure induced by $g_{n-1}$). Let 
+
+	  \end{enumerate}
+	  Before we give the third point, suppose recursively that $g_{n-1}$ already
+	  satisfy all those properties. Let us enumerate 
+	  $\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$
+	  all pairs of finitely generated structures with automorphisms such
+	  that the first substructure embed into the second by inclusion, i.e.
+	  $(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, and $f_{n,k}$
+	  is an embedding of $(A_{n,k}, \alpha_{n,k})$ in the $(\FrGr_{n-1}, g_{n-1})$.
+	  We allow $A_{n,k}$ to be empty. Although $g_{n-1}$ is a finite function,
+	  we may treat it as an automorphism as we have said before.
+
+	  \begin{enumerate}[resume, label=(\roman*)]
+	  \item 
+		Let 
 		$(i, j) = \min\{(\{0, 1, \ldots\} \times \bN) \setminus X_{n-1}\}$ (with the
 		order induced by $\gamma$). Then $X_n = X_{n-1}\cup\{(i,j)\}$ and
 		$(B_{n,k}, \beta_{n,k})$ embeds in $(\FrGr_n, g_n)$ so that this diagram
@@ -178,13 +204,15 @@
 		\end{center}
 	\end{enumerate}
 
-	First item makes sure that no infinite orbit will be present in $g$. The 
-	second item together with the first one are necessary for $g$ to be an 
-	automorphism of $\Gamma$. The third item is the one that gives weak
-	ultrahomogeneity. Now we will show that indeed such $g_n$ may be constructed.
+	% First item makes sure that no infinite orbit will be present in $g$. The 
+	% second item together with the first one are necessary for $g$ to be an 
+	% automorphism of $\Gamma$. The third item is the one that gives weak
+	% ultrahomogeneity. Now we will show that indeed such $g_n$ may be constructed.
 	
-	First, we will suffice the item (iii). Namely, we will construct $\Gamma'_n, g'_n$
-	such that $g_{n-1}\subseteq g'_n$ and $f_{i,j}$ extends to an embedding of
+	First, we will satisfy the item (iii). Namely, we will construct $\Gamma'_n, g'_n$
+	such that $g_{n-1}\subseteq g'_n$, $\Gamma_{n-1}\subseteq\Gamma'_n$,
+	$g'_n$ gives an automorphism of $\Gamma'_n$
+	and $f_{i,j}$ extends to an embedding of
 	$(B_{i,j}, \beta_{i,j})$ to $(\Gamma'_n, g'_n)$. But this can be easily
 	done by the fact, that $\cD$ has the amalgamation property. Moreover, without
 	the loss of generality we can assume that all embeddings are inclusions.
@@ -197,34 +225,48 @@
       \end{tikzcd}
     \end{center}
 
-	By the weak ultrahomogeneity we may assume that $\Gamma'_n\subseteq \Gamma$:
-
-	\begin{center}
-	  \begin{tikzcd}
-		(B_{i,j}\cup\Gamma_{n-1}, \beta_{i,j}\cup g_{n-1}) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Gamma \\
-		(\Gamma'_{n}, g'_n)\arrow[ur, dashed, "f"'] 
-	  \end{tikzcd}
-	\end{center}
-
-	Now, by the WHP of $\cK$ we can extend the graph $\Gamma'_n\cup\{v_n\}$ together
-	with its partial isomorphism $g'_n$ to a graph $\Gamma_n$ together with its
-	automorphism $g_n\supseteq g'_n$ and without the loss of generality we
+	It is important to note that $g'_n$ should be a finite function and once
+	again, as it is an automorphism of a finitely generated structure, we may
+	think it is simply a map from one generators of $\Gamma'_n$ to the 
+	others.	By the weak ultrahomogeneity of $\Gamma$, we may assume that 
+	$\Gamma'_n\subseteq \Gamma$.
+
+	% \begin{center}
+	%   \begin{tikzcd}
+	% 	B_{i,j}\cup\Gamma_{n-1} \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Gamma \\
+	% 	\Gamma'_{n}\arrow[ur, dashed, "f"'] 
+	%   \end{tikzcd}
+	% \end{center}
+
+	Now, by the WHP of $\cK$ we can extend $\langle\Gamma'_n\cup\{v_n\}\rangle$ together
+	with its partial isomorphism $g'_n$ to a finitely generated structure $\Gamma_n$ 
+	together with its
+	automorphism $g_n\supseteq g'_n$ and (again by weak ultrahomogeneity)
+	without the loss of generality we
 	may assume that $\Gamma_n\subseteq\Gamma$. This way we've constructed $g_n$
 	that has all desired properties.
 
 	Now we need to see that $g = \bigcap^{\infty}_{n=0}g_n$ is indeed an automorphism
 	of $\Gamma$ such that $(\Gamma, g)$ has the age $\cH$ and is weakly ultrahomogeneous.
 	It is of course an automorphism of $\Gamma$ as it is defined for every $v\in\Gamma$
-	and is a sum of increasing chain of finite automorphisms.
+	and is an union of an increasing chain of automorphisms of finitely generated
+	substructures.
 
-	Take any finite structure of $T$ with automorphism $(B, \beta)$. Then, there are
+	Take any $(B, \beta)\in\cD$. Then, there are
 	$i, j$ such that $(B, \beta) = (B_{i, j}, \beta_{i,j})$ and $A_{i,j}=\emptyset$.
 	By the bookkeeping there was $n$ such that 
-	$(i, j) = \min\{\{0,1,\ldots\}\times\bN\setminus X_{n-1}\}$.
+	$(i, j) = \min\{\{0,1,\ldots n-1\}\times\bN\setminus X_{n-1}\}$.
 	This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds
-	into $(\Gamma, g)$, thus it has age $\cH$. 
-	With a similar argument we can see that $(\Gamma, g)$ is weakly ultrahomogeneous.
+	into $(\Gamma, g)$. Hence, the age of $(\Gamma, g)$ is $\cH$. 
+
+	It is also weakly ultrahomogeneous. Having $(A,\alpha)\subseteq(B,\beta)$,
+	and an embedding $f\colon(A,\alpha)\to(\Gamma,g)$, we may find $n\in\bN$
+	such that $(i,j) = \min\{\{0,1,\ldots n-1\}\times X_{n-1}\}$ and
+	$(A,\alpha) = (A_{i,j},\alpha_{i,j}), (B,\beta)=(B_{i,j},\beta_{i,j})$ and
+	$f = f_{i,j}$. This means that there is a compatbile embedding of $(B,\beta)$ into
+	$(\Gamma_n, g_n)$, which means we can also embed it into $(\Gamma, g)$.
 
+	Hence, $(\Gamma,g)\cong(\Gamma,\sigma)$.
 	By this we know that $g$ and $\sigma$ conjugate in $G$. As we stated in the
 	beginning of the proof, the set $A$ of possible outcomes of the game (i.e.
 	possible $g$'s we end up with) is comeagre in $G$, thus $\sigma^G$ is also
-- 
cgit v1.2.3


From ae1c456f6467a50427fc485ec5ae163495ea0e52 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Wed, 13 Jul 2022 23:09:49 +0200
Subject: Ortografia

---
 sections/conj_classes.tex | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

(limited to 'sections/conj_classes.tex')

diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index 9620220..c0120d3 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -32,7 +32,7 @@
 	We will show that the conjugacy class of $\sigma$ is an intersection of countably
 	many comeagre sets. 
 
-	Let $A_n = \{\alpha\in Aut(M)\mid \alpha\text{ has infinitely many orbits of size }n\}$.
+	Let $A_n = \{\alpha\in \Aut(M)\mid \alpha\text{ has infinitely many orbits of size }n\}$.
 	This set is comeagre for every $n>0$. Indeed, we can represent this set
 	as an intersection of countable family of open dense sets. Let $B_{n,k}$
 	be the set of all finite functions $\beta\colon M\to M$ that consist
@@ -96,7 +96,7 @@
   $\langle \dom(g)\rangle$ and $\langle\rng(g)\rangle$, i.e. substructures 
   generated by $\dom(g)$ and $\rng(g)$ respectively. Of course, in our
   case, $g$ is good
-  if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (becuase of ultrahomogeneity
+  if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (because of ultrahomogeneity
   of $\Gamma$.
 
   Also it is important to mention that an isomorphism between two finitely
@@ -121,7 +121,7 @@
 	words, a basic open set is a set of all extensions of some finite partial
 	automorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic 
 	open subset given by a finite partial isomorphism $g$. Again, Note that $B_g$
-	is nonemty because of ultrahomogeneity of $\Gamma$.
+	is nonempty because of ultrahomogeneity of $\Gamma$.
 
 	With the use of Corollary \ref{corollary:banach-mazur-basis} we can consider 
 	only games where both players choose finite partial isomorphisms. Namely,
@@ -165,7 +165,7 @@
 	$X_{-1} = \emptyset$. 
 	Suppose that player \textit{I} in the $n$-th move chooses a finite partial
 	automorphism $f_n$. We will construct a finite partial automorphism 
-	$g_n\supseteq f_n$ together with a finitely generated substrucutre
+	$g_n\supseteq f_n$ together with a finitely generated substructure
 	$\Gamma_n \subseteq \Gamma$ and a set $X_n\subseteq\bN^2$ 
 	such that the following properties hold:
 
@@ -263,7 +263,7 @@
 	and an embedding $f\colon(A,\alpha)\to(\Gamma,g)$, we may find $n\in\bN$
 	such that $(i,j) = \min\{\{0,1,\ldots n-1\}\times X_{n-1}\}$ and
 	$(A,\alpha) = (A_{i,j},\alpha_{i,j}), (B,\beta)=(B_{i,j},\beta_{i,j})$ and
-	$f = f_{i,j}$. This means that there is a compatbile embedding of $(B,\beta)$ into
+	$f = f_{i,j}$. This means that there is a compatible embedding of $(B,\beta)$ into
 	$(\Gamma_n, g_n)$, which means we can also embed it into $(\Gamma, g)$.
 
 	Hence, $(\Gamma,g)\cong(\Gamma,\sigma)$.
@@ -290,7 +290,7 @@
   Let $\cC$ be a Fraïssé class of finitely generated $L$-structures with
   weak Hrushovski property and canonical amalgamation. 
   Let $\cD$ be the Fraïssé class (by the Theorem \ref{theorem:key-theorem}
-  of the structures of $\cC$ with additional automorphism of the strucutre.
+  of the structures of $\cC$ with additional automorphism of the structure.
   Let $\Gamma = \Flim(\cC)$. 
 
   \begin{proposition}
-- 
cgit v1.2.3


From 2b830cb2d9c2237fcb7809bab7c64966098ea6fb Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 17 Jul 2022 14:16:41 +0200
Subject: Oby ostatnie poprawki

---
 sections/conj_classes.tex | 72 +++++++++++------------------------------------
 1 file changed, 16 insertions(+), 56 deletions(-)

(limited to 'sections/conj_classes.tex')

diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index c0120d3..9ec4b0c 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -46,7 +46,7 @@
 	$\Aut(M)$: take any finite $\gamma\colon M\to M$ such that $[\gamma]_{\Aut(M)}$
 	is nonempty. Then also 
 	$\bigcup_{\beta\in B_{n,k}} [\beta]_{\Aut(M)}\cap[\gamma]_{\Aut(M)}\neq\emptyset$,
-	one can easily construct a permutation that extends $\gamma$ and have at least
+	one can easily construct a permutation that extends $\gamma$ and has at least
 	$k$ many $n$-cycles.
 
 	Now we see that $A = \bigcap_{n=1}^{\infty} A_n$ is a comeagre set consisting
@@ -97,21 +97,21 @@
   generated by $\dom(g)$ and $\rng(g)$ respectively. Of course, in our
   case, $g$ is good
   if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (because of ultrahomogeneity
-  of $\Gamma$.
+  of $\Gamma$).
 
   Also it is important to mention that an isomorphism between two finitely
   generated structures is uniquely given by a map from generators of one structure
   to the other. This allow us to treat a finite function as an isomorphism
-  of finitely generated structure.
+  of finitely generated structures.
 
   \begin{proof}
 	Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. First, by the Theorem
 	\ref{theorem:isomorphic_fr_lims}, we may assume without the loss of generality
 	that $\Pi = \Gamma$. Let $G = \Aut(\Gamma)$,
 	i.e. $G$ is the automorphism group of $\Gamma$.
-	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 the $\sigma$'s conjugacy class.
+	We will construct a winning strategy for the second player in the Banach-Mazur game
+	(see \ref{definition:banach-mazur-game})
+	on the topological space $G$ with $A$ being $\sigma$'s conjugacy class. 
 	By the Banach-Mazur theorem (see \ref{theorem:banach_mazur_thm}) this will prove 
 	that this class is comeagre.
 
@@ -129,15 +129,6 @@
 	chooses $g_0, g_1,\ldots$ such that
 	$f_0\subseteq g_0\subseteq f_1\subseteq g_1\subseteq\ldots$, which identify
 	the corresponding basic open subsets $B_{f_0}\supseteq B_{g_0}\supseteq\ldots$.
-
-	% Our goal is to choose $g_i$ in such a manner that the resulting function
-	% $g = \bigcap^{\infty}_{i=0}g_i$ will be an automorphism of the Fraïssé limit 
-	% $\Gamma$ such that $(\Gamma, g) = \Flim{\cD}$. 
-	% Precisely, we will find $g_i$'s such that 
-	% $\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$ and by
-	% the Fraïssé theorem \ref{theorem:fraisse_thm}
-	% it will follow that $(\Gamma, g)\cong (\Pi, \sigma)$. Hence,
-	% by the fact	\ref{fact:conjugacy}, $g$ and $\sigma$ conjugate in $G$.
 	
 	Our goal is to choose $g_i$ in such a manner that 
 	$\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$ and $(\Gamma, g)$ is ultrahomogeneous
@@ -145,24 +136,14 @@
 	that $(\Gamma, \sigma)\cong (\Gamma, g)$, thus by the Fact \ref{fact:conjugacy}
 	we have that $\sigma$ and $g$ conjugate.
 
-	% Once again, by the Fraïssé theorem and by Lemma 
-	% \ref{lemma:weak_ultrahom} constructing $g_i$'s in a way such that
-	% age of $(\Gamma, g)$ is exactly $\cD$ and so that it is weakly ultrahomogeneous
-	% will produce expected result. 
-	First, let us enumerate all pairs of structures
-	$\{\langle(A_n, \alpha_n), (B_n, \beta_n)\rangle\}_{n\in\bN},\in\cD$
-	such that the first element of the pair embeds by inclusion in the second,
-	i.e. $(A_n, \alpha_n)\subseteq (B_n, \beta_n)$. Also, it may be that 
-	$A_n$ is an empty. We enumerate the elements of the Fraïssé limit 
-	$\Gamma = \{v_0, v_1, \ldots\}$.
-
 	Fix a bijection $\gamma\colon\bN\times\bN\to\bN$ such that for any
 	$n, m\in\bN$ we have $\gamma(n, m) \ge n$. This bijection naturally induces
 	a well ordering on $\bN\times\bN$. This will prove useful later, as the
 	main ingredient of the proof will be a bookkeeping argument.
 
 	For technical reasons, let $g_{-1} = \emptyset$ and
-	$X_{-1} = \emptyset$. 
+	$X_{-1} = \emptyset$. Enumerate the elements of the Fraïssé limit 
+	$\Gamma = \{v_0, v_1, \ldots\}$.
 	Suppose that player \textit{I} in the $n$-th move chooses a finite partial
 	automorphism $f_n$. We will construct a finite partial automorphism 
 	$g_n\supseteq f_n$ together with a finitely generated substructure
@@ -204,26 +185,13 @@
 		\end{center}
 	\end{enumerate}
 
-	% First item makes sure that no infinite orbit will be present in $g$. The 
-	% second item together with the first one are necessary for $g$ to be an 
-	% automorphism of $\Gamma$. The third item is the one that gives weak
-	% ultrahomogeneity. Now we will show that indeed such $g_n$ may be constructed.
-	
 	First, we will satisfy the item (iii). Namely, we will construct $\Gamma'_n, g'_n$
 	such that $g_{n-1}\subseteq g'_n$, $\Gamma_{n-1}\subseteq\Gamma'_n$,
 	$g'_n$ gives an automorphism of $\Gamma'_n$
 	and $f_{i,j}$ extends to an embedding of
 	$(B_{i,j}, \beta_{i,j})$ to $(\Gamma'_n, g'_n)$. But this can be easily
-	done by the fact, that $\cD$ has the amalgamation property. Moreover, without
-	the loss of generality we can assume that all embeddings are inclusions.
+	done by the fact, that $\cD$ has the amalgamation property. 
 
-    \begin{center}
-      \begin{tikzcd}
-                                  & (\Gamma'_n, g'_n) & \\
-        (B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\subseteq"] &   & (\Gamma_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\
-								  & (A_{i,j}, \alpha_{i,j}) \arrow[ur, "\subseteq"'] \arrow[ul, "\subseteq"]
-      \end{tikzcd}
-    \end{center}
 
 	It is important to note that $g'_n$ should be a finite function and once
 	again, as it is an automorphism of a finitely generated structure, we may
@@ -231,14 +199,7 @@
 	others.	By the weak ultrahomogeneity of $\Gamma$, we may assume that 
 	$\Gamma'_n\subseteq \Gamma$.
 
-	% \begin{center}
-	%   \begin{tikzcd}
-	% 	B_{i,j}\cup\Gamma_{n-1} \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Gamma \\
-	% 	\Gamma'_{n}\arrow[ur, dashed, "f"'] 
-	%   \end{tikzcd}
-	% \end{center}
-
-	Now, by the WHP of $\cK$ we can extend $\langle\Gamma'_n\cup\{v_n\}\rangle$ together
+	Now, by the WHP of $\cC$ we can extend $\langle\Gamma'_n\cup\{v_n\}\rangle$ together
 	with its partial isomorphism $g'_n$ to a finitely generated structure $\Gamma_n$ 
 	together with its
 	automorphism $g_n\supseteq g'_n$ and (again by weak ultrahomogeneity)
@@ -247,7 +208,7 @@
 	that has all desired properties.
 
 	Now we need to see that $g = \bigcap^{\infty}_{n=0}g_n$ is indeed an automorphism
-	of $\Gamma$ such that $(\Gamma, g)$ has the age $\cH$ and is weakly ultrahomogeneous.
+	of $\Gamma$ such that $(\Gamma, g)$ has the age $\cD$ and is weakly ultrahomogeneous.
 	It is of course an automorphism of $\Gamma$ as it is defined for every $v\in\Gamma$
 	and is an union of an increasing chain of automorphisms of finitely generated
 	substructures.
@@ -257,7 +218,8 @@
 	By the bookkeeping there was $n$ such that 
 	$(i, j) = \min\{\{0,1,\ldots n-1\}\times\bN\setminus X_{n-1}\}$.
 	This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds
-	into $(\Gamma, g)$. Hence, the age of $(\Gamma, g)$ is $\cH$. 
+	into $(\Gamma, g)$. Thus, $\cD$ is a subclass of the age of $(\Gamma, g)$.
+	The other inclusion is obvious.	Hence, the age of $(\Gamma, g)$ is $\cH$. 
 
 	It is also weakly ultrahomogeneous. Having $(A,\alpha)\subseteq(B,\beta)$,
 	and an embedding $f\colon(A,\alpha)\to(\Gamma,g)$, we may find $n\in\bN$
@@ -267,11 +229,9 @@
 	$(\Gamma_n, g_n)$, which means we can also embed it into $(\Gamma, g)$.
 
 	Hence, $(\Gamma,g)\cong(\Gamma,\sigma)$.
-	By this we know that $g$ and $\sigma$ conjugate in $G$. As we stated in the
-	beginning of the proof, the set $A$ of possible outcomes of the game (i.e.
-	possible $g$'s we end up with) is comeagre in $G$, thus $\sigma^G$ is also
-	comeagre and $\sigma$ is a generic automorphism, as it contains a comeagre
-	set $A$.
+	By this we know that $g$ and $\sigma$ are conjugate in $G$, thus player \textit{II}
+	have a winning strategy in the Banach-Mazur game with $A=\sigma^G$,
+	thus $\sigma^G$ is comeagre in $G$ and $\sigma$ is a generic automorphism.
   \end{proof}
 
   \begin{theorem}
-- 
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