# Definition:Saturation (Equivalence Relation)

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## Contents

## Definition

Let $\sim$ be an equivalence relation on a set $S$.

Let $T\subset S$ be a subset.

### Definition 1

The **saturation of $T$** is the set of all elements that are equivalent to some element in $T$:

- $\overline T = \{s \in S : \exists t\in T : s\sim t\}$

### Definition 2

The **saturation of $T$** is the union of the equivalence classes of its elements:

- $\displaystyle \overline T = \bigcup_{t \mathop \in T} \eqclass t \sim$

### Definition 3

The **saturation of $T$** is the preimage of its image under the quotient mapping:

- $\overline T = q^{-1} \sqbrk {q \sqbrk T}$

## Also denoted as

The saturation is also denoted $\operatorname{Sat}(T)$.

## Also see

- Equivalence of Definitions of Saturation Under Equivalence Relation
- Definition:Saturated Set (Equivalence Relation)
- Definition:Saturation (Group Action)
- Saturation Under Equivalence Relation in Terms of Graph, an equivalent definition of saturation that is useful in topology