Definition. Let $X$ and $Y$ be topological spaces; let $p : X \to Y$ 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 $X$.
This condition is stronger than continuity; some mathematicians call it "strong continuity." An equivalent condition is to require that a subset $A$ of $Y$ be closed in $Y$ if and only if $p^{-1}(A)$ is closed in $X$. Equivalence of the two conditions follows from equation $$f^{-1}(Y-B)=X-f^{-1}(B).$$
This is quoted from Munkres' Topology. The sentence that I don't understand is that "An equivalent condition is to require that a subset $A$ of $Y$ be closed in $Y$ if and only if $p^{-1}(A)$ is closed in $X$."
I want to understand what equivalent condition does this sentence discuss? Does it mean if a map $p$ satisfies the condition that a subset $A$ of $Y$ be closed in $Y$ if and only if $p^{-1}(A)$ is closed in $X$, then it is a quotient map?
Yes. Assuming the second statement holds, then given open set $U$ of $Y$, then $Y-U$ is closed in Y, then $p^{-1}(Y-U)$ is closed in $X$, but $p^{-1}(Y-U)=X-p^{-1}(U)$, so $p^{-1}(U)$ is open in $X$.
Assume that $p^{-1}(U)$ is open in $X$, then $X-p^{-1}(U)$ is closed in $X$, as in before, $p^{-1}(Y-U)$ is closed in $X$, so $Y-U$ is closed in $Y$, and finally $U$ is open in $Y$.
This shows that $p$ is a quotient map.
By the similar reasoning, by assuming the original definition of quotient map, one proves the second statement.