From e48745a4ecae399b000828b02ff69ec3ea4febef Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Wed, 16 Feb 2022 14:06:29 +0100 Subject: Corollary 2.15 --- lic_malinka.pdf | Bin 316851 -> 317628 bytes lic_malinka.tex | 14 ++++++++++++-- 2 files changed, 12 insertions(+), 2 deletions(-) diff --git a/lic_malinka.pdf b/lic_malinka.pdf index b58ba11..45f4de6 100644 Binary files a/lic_malinka.pdf and b/lic_malinka.pdf differ 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} -- cgit v1.2.3