Defining the quotient map

Question

In Introduction to Topological Manifolds by John Lee the quotient map is defined in the following way

Let $ q : X \to X/$~ be the natural projection sending each element of $X$ to its equivalence class, then $X/$~ together with the quotient topology induced by $q$ is called the quotient space

where $X$ is a topological space and $X/$~ is the quotient space. My question is that the above definition implies $q(x) = [x]$, hence it seems like $q$ isn't even a function as it may map a single point $x$ to many points in the equivalence class $[x]$

Is there an error in my understanding?

Answer 1

If $x,y$ belong to the same equivalence class then

$$ [x]=[y] $$

you are free to use the symbol $[x]$ or $[y]$ to represent the same equivalence class.

Your function $q$ will not be multivalued since

$$ q(x)=[x] $$

$$ q(x)=[y] $$

but as stated previously $[x]$ and $[y]$ are the same element.

Answer 2

A common way to think of quotient maps is that they take each equivalence class and shrink it to a single point. That single point is the image, under $q()$ of every point that was in the equivalence class.

For a simple, concrete example, consider the function $f:\mathbb{C}\to\mathbb{C}$ defined by $f(z)=e^{iz}$, and consider its restriction to the line segment $[0,2\pi]$. This maps sends a line segment to a circle, and in doing so, it sends the points $0$ and $2\pi$ to the same point, namely $1$. The equivalence classes of this quotient are singletons for every point in the open interval $(0,2\pi)$, and the two-element equivalence class $\{0,2\pi\}$, which we could call $[0]$ or $[2\pi]$.

We can think of $f$ as a composition of two maps: $f=\iota\circ q$, where $q$ is a purely topological quotient map which sends each point to the singleton consisting of itself, except for the end points, which are "glued" together in a single equivalence class, and $\iota$ is an inclusion map which sticks the resulting circle back into $\mathbb{C}$.

It should be clear that $f(0) = f(2\pi) = 1 =\iota(\{0,2\pi\})$, so both points are sent to a single element of the image, even though you can see that each of those points is mapped to the common destination of two points. Thus, a quotient map is not a "one-to-many" map, which would not be a function, but it is a "many-to-one" map, i.e., a non-injective function.