Def. for a continuous function:
Let $X$ and $Y$ be topological spaces. A function $f : X \rightarrow Y$ is continuous if $f^{-1} (V)$ is open in $X$ for every open set $V \subseteq Y$.
Def. for a quotient map:
Let $X$ be a topological space and $A$ be a set (that is not necessarily a subset of $X$). Let $p : X \rightarrow A$ be a surjective map. Define a subset $U$ of $A$ to be open in $A$ if and only if $p^{-1} (U)$ is open in $X$. The resultant collection of open sets in $A$ is called the quotient topology induced by $p$ , and the function $p$ is called a quotient map.
In addition to surjectivity of a quotient map, are there any differences between this two concepts? From many different examples in comparison, they seem to be different 'topics' but actually no other difference more than surjectivity in case of a quotient map (?)
Also the book claims that a quotient map is always a continuous function.