In my experience the usual context in which saturated sets appear in topology is the one metioned by Stefan H. in the comments: you have a map $f:X\to Y$, and you say that a set $A\subseteq X$ is saturated with respect to $f$ iff $A=f^{-1}\big[f[A]\big]$. More generally, if $\mathscr{P}$ is a partition of $X$, a set $A\subseteq X$ is saturated with respect to $\mathscr{P}$ iff $A=\bigcup\{P\in\mathscr{P}:P\cap A\ne\varnothing\}$. (This really is a generalization: in the case of the map $f$, the associated partition of $X$ is $\{f^{-1}[\{y\}]:y\in Y\}$, the set of fibres of the map $f$.)
If $\mathscr{P}$ is a partition of $X$ and $A$ is an arbitrary subset of $X$, there are two saturated sets naturally associated with $A$. One, which we might call the outer saturation of $A$, is $\bigcup\{P\in\mathscr{P}:P\cap A\ne\varnothing\}\;;$ if $\mathscr{P}$ is generated as above by a map $f:X\to Y$, the outer saturation of $A$ is $A=f^{-1}\big[f[A]\big]$. The other, which we might call the inner saturation of $A$, is $\bigcup\{P\in\mathscr{P}:P\subseteq A\}\;;$ it’s the complement of the outer saturation of $X\setminus A$, so if $\mathscr{P}$ is generated as above by a map $f:X\to Y$, the inner saturation of $A$ is $X\setminus f^{-1}\big[f[X\setminus A]\big]$.
My best guess without having seen the actual context is that by the saturation of a set $A$ they mean what I’ve called here the outer saturation of $A$.
Note: The terms outer saturation and inner saturation are not standard, so far as I know; I’m using them here for purposes of exposition.