I am having a little difficulty in understanding the idea of a saturated set and quotient maps. My book says
We say that a subset $C$ of is saturated (with respect to the surjective map $p : X \rightarrow Y$ if $C$ contains set $p^{-1}(\{y\})$ that it intersects
Is this saying that $p^{-1}(\{y\}) \cap C \neq \emptyset$ implies $p^{-1}(\{y\}) \subseteq C$? If so, then it seems that $\bigcup_{y \in D} p^{-1}(\{y\}) \subseteq C$, where $D = \{y \in Y ~|~ p^{-1}(\{y\}) \cap C \neq \emptyset \}$; or $p^{-1}(D) \subseteq C$, since $\bigcup_{y \in D} p^{-1}(\{y\}) = p^{-1}(\bigcup_{y \in D} \{y\}) = p^{-1}(D)$. Is this right?
It then goes on to say
Thus $C$ is saturated if it equals the complete inverse image of a subset of $Y$.
Is this saying that $p^{-1}(D) = C$? Why wouldn't this be an "if and only if"? I find this somewhat odd. Why isn't this true of any set? For instance, if $p : \mathbb{R} \rightarrow \mathbb{R}$ with $p(x) = x^2$ (which isn't surjective, mind you), then if $C=[-1,1]$, we have $p^{-1}([0,1]) = [-1,1] = C$, showing that $C$ equals the complete inverse image of the subset $[0,1]$ of $Y = \mathbb{R}$? What am I misunderstanding?
Three equivalent statements concerning a function $p:X\to Y$ and a set $C\subseteq X$ are:
Iff these statements are true then $C$ is saturated wrt $p$. I leave open which of the statements is most suitable to be used as definition. Personally I would go for the second.
Actually a saturated set wrt $p$ can be looked at as a union of $p$-fibers and in the context of surjective maps these fibers correspond one to one with the elements of $Y$.
To get some understanding of saturated sets it is a good exercise to prove the equivalences.