From 30e20714fa82c6d0d6b1c06b81ebcefdb72e1004 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 10 Jul 2022 19:24:51 +0200
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 =?UTF-8?q?am=20zmiany?=
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---
 sections/preliminaries.tex | 16 ++++++++++++++++
 1 file changed, 16 insertions(+)

(limited to 'sections/preliminaries.tex')

diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex
index 729ef1d..266845c 100644
--- a/sections/preliminaries.tex
+++ b/sections/preliminaries.tex
@@ -52,6 +52,22 @@
     }x\}$ is comeagre in $X$.  
   \end{definition}
 
+  Let $M$ be a structure. We define a topology on the automorphism group 
+  $\Aut(M)$ of $M$ by the basis of open sets: for a finite function
+  $f\colon M\to M$ we have a basic open set 
+  $[f]_{\Aut(M)} = \{g\in\Aut(M)\mid f\subseteq g\}$. This is a standard
+  definition.
+
+  \begin{fact}
+	For a countable structure $M$, the topological space $\Aut(M)$ is a 
+	Baire space.
+  \end{fact}
+
+  This is in fact a very weak statement, it is also true that $\Aut(M)$ is
+  a Polish space (i.e. separable completely metrizable), and every Polish
+  space is Baire. However, those additional properties are not important in
+  this study.
+
   \begin{definition}
 	\label{definition:generic_automorphism}
 	Let $G = \Aut(M)$ be the automorphism group of structure $M$. We say
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