What is a saturation of a set?

So, I encountered in my topology book the saturation of a set and In my first language the translated word is rarely used and those papers I found who use it don't explain it since it seems to be something very basic. Unfortunately the part where the saturation is mentioned doesn't really give many information about what it could mean.

My Question: When do we call a set saturated? (I guess I understand it to a certain extend) What is the saturation of a set (which is not saturated)?

I hope someone can explain it to me. The corresponding wikipedia entry http://en.wikipedia.org/wiki/Saturated_set only explains when you call a subset of a topoligical space saturated but doesn't explain how to achieve the saturation of a set which isn't saturated.

Solution 1:

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.