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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?

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

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    $\begingroup$ I would call it "the coarsest equivalence relation with which every set in $\mathcal{F}$ is compatible". $\endgroup$
    – Nik Weaver
    Commented Oct 23, 2019 at 18:29
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    $\begingroup$ "The partition generated by $\mathcal{F}$"? $\endgroup$
    – YCor
    Commented Oct 23, 2019 at 21:45

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$\mathscr F$-indistinguishability.

In analogy with Topological indistinguishability.

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The elements of this partition are precisely the atoms of the complete Boolean algebra generated by the family.

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    $\begingroup$ Which is atomic, so the equivalence classes are the atoms. $\endgroup$
    – François G. Dorais
    Commented Oct 23, 2019 at 22:36

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