Given two topological spaces $X$ and $Y$, we define the map $f$ from $X$ to $Y$ as perfect map if $f$ is continuous, surjective, closed, and proper map (or that inverse image of each point in $Y$ is compact in $X$).
My question is that is the requirement of surjectivity really needed? I mean even if a function is not surjective, the inverse image of unmapped point $y$ in $Y$ is empty set, which is compact. So why would surjectivity required as a definition?
The point of requiring it to be surjective is that we typically want to use it and properties of $X$ to tell us something about $Y$, not just about $f[X]$. For instance, perfect maps preserve regularity, local compactness, and second countability. They are also quotient maps, so surjectivity means that $Y$ is a quotient of $X$.