Saturated set

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.