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 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.