Understanding quotient map

If I am understanding correctly, a quotient map can be defined in this way (actually I quoted the following from Munkres):

Let $X$ and $Y$ be topological spaces; let $p:X \rightarrow Y$ be a surjective map. The map $p$ is said to be a quotient map provided a subset $U$ of $Y$ is open in $Y$ if and only if $p^{-1}(U)$ is open in $X$.

Or in this way:

Let $C$ be an equivalence relation in $X \times X$. The map $p:X \rightarrow X/C$ s.t. $p(x)=[x]$ ($x \in X, [x]$ is an equivalence class) is defined to be a quotient map.

I understood the concept from the second one, since equivalence classes partition a set, which is very easy to visualize. (Can anyone answer this?) My lecturer introduced equivalence relation and quotient space first also. Only when I need the strong continuity I will consider the first one, which is convenient. However when I read Munkres, I have no idea why those two describe the same thing (at least from my opinion). Can someone explain why to me?

1 Answer

Note that you didn't provide a topology in the second case. Actually it is defined the same way as in the first case, that $U\in X/C$ is open iff $p^{-1}(U)$ is open.

And in the first case, a surjective map naturally induces an equivalence relation on $X$ with $x_1\sim x_2 \Leftrightarrow f(x_1)=f(x_2)$ with exactly $Y$ the set of equivalence classes.