Definition:Saturated Set (Equivalence Relation)
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Definition
Let $\sim$ be an equivalence relation on a set $S$.
Let $T\subset S$ be a subset.
Definition 1
$T$ is saturated if and only if it equals its saturation:
- $T = \overline T$
Definition 2
$T$ is saturated if and only if it is a union of equivalence classes:
- $\ds \exists U \subset S : T = \bigcup_{u \mathop \in U} \eqclass u {}$
Definition 3
$T$ is saturated if and only if it is the preimage of some set under the quotient mapping:
- $\exists V \subset S / \sim \; : T = q^{-1} \left[{V}\right]$