Fiber (mathematics)

In mathematics, the fiber (US English) or fibre (British English) of an element ${\displaystyle y}$ under a function ${\displaystyle f}$ is the preimage of the singleton set ${\displaystyle \{y\}}$,^[1]: p.69 that is

${\displaystyle f^{-1}(\{y\})=\{x\mathrel {:} f(x)=y\}}$

As an example of abuse of notation, this set is often denoted as ${\displaystyle f^{-1}(y)}$, which is technically incorrect since the inverse relation ${\displaystyle f^{-1}}$ of ${\displaystyle f}$ is not necessarily a function.