Ultrametric space of stochastic filtration

Question

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that is $\lim_{t\to\infty}\mathscr F_t = 2^\Omega$. Define $$ d(A,B):=\inf\{t\in \Bbb R_+: A\Delta B\in \mathscr F_t\}\qquad A,B\subseteq \Omega. $$ where $\Delta$ is a symmetric difference of sets. One can show that $d$ is a ultra-metric: a function that satisfies $d(A,C) \leq\max(d(A,B),d(B,C))$ instead of a triangular inequality. In fact it is a pseudo ultra-metric sicen $d(A,B) = 0$ does not imply $A = B$. Was such ultra-metric studied somewhere?