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Diffstat (limited to 'sections/conj_classes.tex')
| -rw-r--r-- | sections/conj_classes.tex | 12 |
1 files changed, 6 insertions, 6 deletions
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} |