2013 seminar talk: Polish G-spaces similar to logic G-spaces of continuous structures

Talk held by Aleksander Ivanov and Barbara Majcher-Iwanow (University of Wrocław, Poland) at the KGRC seminar on 2013-01-07.


We extend the concept of nice topologies of H.Becker to the general case of Polish $G$-spaces (Becker assumed that $G\lt Sym(\omega)$). Our apprach is based on continuous first order logic.

Let $({\bf Y},d)$ be a Polish space and $Iso({\bf Y},d)$ be the corresponding isometry group endowed with the pointwise convergence topology. Then $Iso ({\bf Y},d)$ is a Polish group. It is worth noting that any Polish group $G$ can be realised as a closed subgroup of the isometry group $Iso ({\bf Y},d)$ of an appropriate Polish space ${\bf Y}$.

For any countable continuous signature $L$ the set ${\bf Y}_L$ of all continuous metric $L$-structures on $({\bf Y},d)$ can be considered as a Polish $Iso({\bf Y},d)$-space. We call this action {\em logic}. Note that for any tuple $\bar{s}\in {\bf Y}$ the map $g\rightarrow d(\bar{s},g(\bar{s}))$ can be considered as a graded subgroup of $Iso({\bf Y},d)$. For any continuous sentence $\phi$ we have a graded subset of ${\bf Y}_L$ defined by $M \rightarrow \phi^{M}$.

We investigate Polish $G$-spaces ${\bf X}$ where $G$ is Polish. We pove that distinguishing an appropriate family of graded subgroups of $G$ and some family $\mathcal{B}$ of graded subsets of ${\bf X}$ (called a graded nice basis) we arrive at the situation very similar to the logic space $\mathcal{U}_L$, where $\mathcal{U}$ is the bounded Urysohn space. Treating elements of $\mathcal{B}$ as continuos formulas we obtain topological generalisations of several theorems from logic, for example Ryll-Nardzewski theorem.

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