Quasi-metrizability of bornological biuniverses in ZF (Q2789234)

A bornology \(\mathcal B\) on a set \(X\) is an ideal on \(X\) containing every singleton of \(X\). A bornological biuniverse is a pair consisting of a bitopological space and a bornology on the underlying set. A bornological biuniverse \(((X,\tau_1,\tau_2),\mathcal B)\) is (quasi-)metrisable if there is a (quasi-)metric \(d\) on \(X\) such that \(\tau_1\) and \(\tau_2\) are the topologies induced by \(d\) and its conjugate and \(\mathcal B\) consists precisely of the subsets of \(X\) bounded under \(d\), while \(\mathcal B\) is \((\tau_1,\tau_2)\)-proper if for each \(a\in\mathcal B\) there is \(B\in\mathcal B\) such that \(\text{cl}_{\tau_2}A\subset\text{int}_{\tau_1}B\). A slightly strengthened form of the following \textbf{ZF} result is shown: \(((X,\tau_1,\tau_2),\mathcal B)\) is quasi-metrisable if and only if \((X,\tau_1,\tau_2)\) is quasi-metrisable and \(\mathcal B\) has a countable base and is \((\tau_1,\tau_2)\)-proper. Other results relating to versions of metrisability of a bornological universe using only \textbf{ZF}, possibly plus a weakened form of \textbf{AC}, are presented.