The fiber of an element $y\in Y$ (or fiber over$y$) under the map $f$ is the set

$f^{-1}(y)=\{x\in X:f(x)=y\},$

that is, the set of elements that get mapped to $y$ by the function. It is the
preimage of the singleton $\{y\}.$ (One usually takes $y$ in the
image of $f$ to avoid $f^{-1}(y)$ being the
empty set.)

The collection of all fibers for the function $f$ forms a
partition of the domain $X.$ The fiber containing an element $x\in X$ is the set $f^{-1}(f(x)).$ For example, the fibers of the projection map $\mathbb {R} ^{2}\to \mathbb {R}$ that sends $(x,y)$ to $x$ are the vertical lines, which form a partition of the plane.

A function between topological spaces is called monotone if every fiber is a
connectedsubspace of its domain. A function $f:\mathbb {R} \to \mathbb {R}$ is monotone in this topological sense if and only if it is
non-increasing or
non-decreasing, which is the usual meaning of "
monotone function" in
real analysis.

A function between topological spaces is (sometimes) called a proper map if every fiber is a
compact subspace of its domain. However, many authors use other non-equivalent competing definitions of "proper map" so it is advisable to always check how a particular author defines this term.
A
continuousclosedsurjective function whose fibers are all compact is called a perfect map.