Relation between partitions and saturation (quotient spaces) A lot of books seem to define quotient topologies in terms of equivalence classes and partitions of the space. However, Munkres does not. He defines the quotient map in terms strong continuity. Also, he defines it in terms of saturated sets. That is, a continuous map is a quotient map if it maps saturated open sets to open sets.
The definition of saturation is, according to Munkres, that a subset C of X is saturated if it contains every $p^{-1}(y)$ that is intersects. However, I have also seen a saturated set being defined as the union of elements of a partition (Elementary Topology by Viro et al.).
So, my questions is, what is the relation between the concepts or partitions and saturation? It seems intuitive that partitions will give you saturated sets. However, does every continuous saturated map induce a partition?
The main question here is, are these different definitions of a quotient space equivalent? If yes, where can I find a proof? If no, what is a counter-example?
 A: Let $X$ and $Y$ be sets; then any surjection $f:X\to Y$ induces a partition of $X$, namely,
$$\left\{f^{-1}[\{y\}]:y\in Y\right\}\;.$$
The elements of the partition are the fibres (point inverses) of the function $f$. The only reason that I specified that $f$ was surjective is to ensure that all of the fibres are non-empty, since technically a partition is not allowed to have empty parts. If $f$ is not surjective, it still induces a partition of $X$ into its non-empty fibres.
This is true without any reference to topology, and – still without reference to topology – you can talk about saturated subsets of $X$: they are the sets that are unions of fibres of $f$. This is exactly the same as saying that they are the sets $A\subseteq X$ such that if $A\cap f^{-1}[\{y\}]\ne\varnothing$ for some $y\in Y$, then $f^{-1}[\{y\}]\subseteq A$.
Of course the partition into $f$-fibres is also associated with an equivalence relation: $x_0,x_1\in X$ are related if and only if $f(x_0)=f(x_1)$.
A: A quotient map induce a partition in it's domain.The elements of partion are the saturated sets.
Now, note that a partition of the domain give you a equivalece relation, then the projection is a quotient map.
