Dynamical pair assignments

Relations between points in the phase space are central to the study of topological dynamical systems. Since many of these relations share common properties, it is natural to study them within a unified framework. To this end, we introduce the concept of $\mathit{\text{dynamical}}\mathit{\text{pair}}\mathit{\text{assignments}}$ $\mathcal{P}$. We then introduce the notions of a dynamical system being $\mathcal{P}$-full and $\mathcal{P}$-realizable, which generalize several existing concepts in the field like CPE, weak mixing and UPE. Our results establish that the space of $\mathcal{P}$-full systems is always a Borel set, while the space of $\mathcal{P}$-realizable systems is Borel if and only if an associated natural rank is bounded.