Corollary 2.15

1 files changed, 12 insertions, 2 deletions

diff --git a/lic_malinka.tex b/lic_malinka.tex
index 0b03eb7..62d1f7b 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -255,15 +255,25 @@
     %     \item rozwinąć \ref{lemma:comprehensive_lemma} (ii), poza tym $W_{n+1}$ jest dokładnie równy, a nie tylko superset, dodać kwantyfikator na n, a może wplecić jeszcze to że te rodziny $W_n$ też są rozłączne?

     % \end{itemize}

 

+    \begin{corollary}

+        \label{corollary:banach-mazur-basis}

+        If we add a constraint to the Banach-Mazur game such that players can only choose basic open sets, then the theorem \ref{theorem:banach_mazur_thm} still suffices.

+    \end{corollary}

+

+    \begin{proof}

+        If one adds the word \textit{basic} before each occurance of word \textit{open} in previous proofs and theorems then they all will still be valid (except for $\Rightarrow$, but its an easy fix -- take $V_n$ a basic open subset of $U_n\cap A_n$).  

+    \end{proof}

+

+    This corollary will be important in using the theorem in practice -- it's much easier to work with basic open sets rather than any open sets.

+

     \subsection{Fraïssé classes}

     \begin{fact}[Fraïssé theorem]

         \label{fact:fraisse_thm}

         % Suppose $\cC$ is a class of finitely generated $L$-structures such that...

 

-        Then there exists a unique up to isomorphism counable $L$-structure $M$ such that...

+        Then there exists a unique up to isomorphism countable $L$-structure $M$ such that...

     \end{fact}

 

-

     \begin{definition}

         For $\cC$, $M$ as in Fact~\ref{fact:fraisse_thm}, we write $\Flim(\cC)\coloneqq M$.

     \end{definition}