# Definition:Saturated Set (Equivalence Relation)

## 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:

$\displaystyle \exists U \subset S : T = \bigcup_{u \mathop \in U} \left[\!\left[{u}\right]\!\right]$

### Definition 3

$T$ is saturated if and only if it is the preimage of some set under the quotient mapping:

$\exists V \subset S / \sim \; : T = q^{-1} \left[{V}\right]$