# Is there a name for this equivalence relation?

Let $$M$$ be an arbitrary set and let $$\mathscr{F}$$ be a family of subsets of $$M$$. Is there a known name for the following equivalence relation or its corresponding partition?

$$\sim_{M,\mathscr{F}}\,=\bigl\{(x,y)\in M\times M\bigm|\forall A\in\mathscr{F}\,(x\in A\leftrightarrow y\in A)\bigr\}$$.

• I would call it "the coarsest equivalence relation with which every set in $\mathcal{F}$ is compatible". – Nik Weaver Oct 23 at 18:29
• "The partition generated by $\mathcal{F}$"? – YCor Oct 23 at 21:45

$$\mathscr F$$-indistinguishability.