In mathematics, particularly in the subfields of set theory and topology, a set ${\displaystyle C}$ is said to be saturated with respect to a function ${\displaystyle f:X\to Y}$ if ${\displaystyle C}$ is a subset of ${\displaystyle f}$'s domain ${\displaystyle X}$ and if whenever ${\displaystyle f}$ sends two points ${\displaystyle c\in C}$ and ${\displaystyle x\in X}$ to the same value then ${\displaystyle x}$ belongs to ${\displaystyle C}$ (that is, if ${\displaystyle f(x)=f(c)}$ then ${\displaystyle x\in C}$). Said more succinctly, the set ${\displaystyle C}$ is called saturated if ${\displaystyle C=f^{-1}(f(C)).}$

In topology, a subset of a topological space ${\displaystyle (X,\tau )}$ is saturated if it is equal to an intersection of open subsets of ${\displaystyle X.}$ In a T₁ space every set is saturated.

Let ${\displaystyle f:X\to Y}$ be a map. Given any subset ${\displaystyle S\subseteq X,}$ define its image under ${\displaystyle f}$ to be the set: $${\displaystyle f(S):=\{f(s)~:~s\in S\}}$$ and define its preimage or inverse image under ${\displaystyle f}$ to be the set: $${\displaystyle f^{-1}(S):=\{x\in X~:~f(x)\in S\}.}$$

Given ${\displaystyle y\in Y,}$ the fiber of ${\displaystyle f}$ over ${\displaystyle y}$ is defined to be the preimage: $${\displaystyle f^{-1}(y):=f^{-1}(\{y\})=\{x\in X~:~f(x)=y\}.}$$

Any preimage of a single point in ${\displaystyle f}$'s codomain ${\displaystyle Y}$ is referred to as a fiber of ${\displaystyle f.}$

A set ${\displaystyle C}$ is called ${\displaystyle f}$-saturated and is said to be saturated with respect to ${\displaystyle f}$ if ${\displaystyle C}$ is a subset of ${\displaystyle f}$'s domain ${\displaystyle X}$ and if any of the following equivalent conditions are satisfied:

Related to computability theory, this notion can be extended to programs. Here, considering a subset ${\displaystyle A\subseteq \mathbb {N} }$, this can be considered saturated (or extensional) if ${\displaystyle \forall m,n\in \mathbb {N} ,m\in A,\vee \phi _{m}=\phi _{n}\Rightarrow n\in A}$. In words, given two programs, if the first program is in the set of programs satisfying the property and two programs are computing the same thing, then also the second program satisfies the property. This means that if one program with a certain property is in the set, all programs computing the same function must also be in the set).

In this context, this notion can extend Rice's theorem, stating that: