Definition:Finer Equivalence Relation

Definition

Let $X$ be a set.

Let $\equiv$ and $\sim$ be equivalence relations on $X$.

Then $\equiv$ is finer than $\sim$ if and only if:

$\forall x, y \in X : x \equiv y \implies x \sim y$

Also known as

If $\equiv$ is finer than $\sim$, then $\sim$ is said to be coarser than $\equiv$.

Also see

• Equivalent Characterizations of Finer Equivalence Relation

Sources

• 1986: Nicolas Bourbaki: Theory of Sets ... (next) Chapter $\text I$: Description of Formal Mathematics: $\S 6$ Equivalence relations $7$. Quotients of equivalence relations