simpler and more natural objects from the point of view of model theory. This motivates our interest in the category of definable $G$-flows. Another, more concrete motivation stems from the fact that even if the language and the model $M$ are both countable, the universal externally definable $G$-ambit $S_{G,\operatorname{{ext}}}(M)$ may be “big” (e.g. not metrizable) which causes some difficulties in the application of topological dynamics to Borel cardinalities of bounded invariant equivalence relations [@KrPiRz]. In contrast, as we explain in this paper, under such a countability assumption, the universal definable $G$-ambit is always metrizable, which may lead to simplifications of some proofs concerning Borel cardinalities or even to new results. On the other hand, our research leads to interesting questions and relations with Polish structures introduced in [@Kr] and further developed by several authors, e.g. in [@KrWa].

In this paper, we will consider even a more general category than that of definable $G$-flows, namely the category of $\Delta$-definable $G$-flows, where $\Delta$ is an arbitrary set of formulas $\delta(x;y,z,\bar t)$ of the form $\varphi(y\cdot x \cdot z, \bar t)$ which contains a formula defining $G$. The definitions of $\Delta$-formulas, the space $S_{G,\Delta}(M)$, and $\Delta$-definable $G$-flows are given in Section \[section: preliminaries\].

