From e48745a4ecae399b000828b02ff69ec3ea4febef Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Wed, 16 Feb 2022 14:06:29 +0100
Subject: Corollary 2.15

---
 lic_malinka.tex | 14 ++++++++++++--
 1 file changed, 12 insertions(+), 2 deletions(-)

(limited to 'lic_malinka.tex')

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}
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