Transfinite sequences of topologies, descriptive complexity, and approximating equivalence relations (Q2040222)

An increasing transfinite sequence \((\tau_\xi)_{\xi<\rho}\) of topologies weaker than \(\tau\) on \(X\) is a ``filtration from \(\sigma\) to \(\tau\)'' if \(\tau_0=\sigma\) and the \(\tau_\alpha\)-interior and \(\tau\)-interior of \(\tau_\xi\)-closed sets \(F\) coincide for every \(\xi<\alpha<\rho\). The main results assume that \(\tau\) is a regular and Baire topology on \(X\) and the topologies \(\tau_\xi\) of the filtration are completely metrizable. Under the additional assumption that \(\tau\) has a neighbourhood basis consisting of \(C\)-sets in \(\sigma\), the filtrations stabilize by reaching \(\tau\). More precisely, if \(\tau_{\xi_0}=\tau_{\xi_0+1}\), then \(\tau_{\xi_0}=\tau\). Under the stronger additional assumption that \(\tau\) has a neighbourhood basis consisting of \(\sigma\)-Borel sets, the filtrations from \(\sigma\) to \(\tau\) with \(\rho\ge\omega_1\) stabilize by \(\tau_{\omega_1}=\tau\). Additional assumptions on the existence of a neighbourhood basis of \(\tau\) consisting of sets of some concrete Borel classes related to \(\alpha<\omega_1\) ensure that \(\tau\) is reached at the latest by \(\tau_\alpha\). A natural sequence of equivalence relations \(E_\xi\) is related to each filtration \((\tau_\xi)_{\xi<\rho}\) by making equivalent those \(x,y\in X\) for which the \(\tau_\xi\)-closures of the equivalence classes which contain \(x\) or \(y\) coincide. Let us formulate a particular corollary of the main result under the common assumptions on the filtration mentioned above. If the equivalence classes of some equivalence relation \(E\) are \(\tau\)-open and \(\sigma\)-Borel, then \(\bigcap_{\xi<\omega_1}E_\xi=E\).