The book I'm reading isn't very clear and doesn't provide any concrete examples or definitions as to what a quotient topology is besides this paragraph.
Let $(X,T)$ be any topological space and ~ any equivalence relation on $X$. Let $Y$ be the set of all equivalence classes of ~. We can denote $Y$ by $X/$~. The natural topology to put on the set $Y = X/$~ is the quotient toplogy under the map which identifies the equivalence classes; that is, maps each equivalence class to a point.
I don't understand this, what exactly is a quotient topology? What does it mean by a map that identifies equivalence classes? If a function maps an equivalence class to a point, then isn't it a function from $Y$ to $X$?