From 30e20714fa82c6d0d6b1c06b81ebcefdb72e1004 Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Sun, 10 Jul 2022 19:24:51 +0200 Subject: =?UTF-8?q?Dodany=20wst=C4=99p=20po=20polsku=20i=20jakie=C5=9B=20t?= =?UTF-8?q?am=20zmiany?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 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 -- cgit v1.2.3