Defining the quotient map 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? 

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.
