fibre

Given a function $f:X⟶Y$, a fibre is an inverse image of an element of $Y$. That is given $y\in Y$, $f^{-1}(\{y\})=\{x\in X\mid f(x)=y\}$ is a fibre.

Example:Define $f:{ℝ}^2⟶ℝ$ by $f(x,y)=x^2+y^2$. Then the fibres of $f$ consist of concentric circles about the origin, the origin itself, and empty sets depending on whether we look at the inverse image of a positive number, zero, or a negative number respectively.

Example:Suppose $M$ is a manifold, and $\pi :TM\to M$ is thecanonical projection from the tangent bundle $TM$ to $M$. Thenfibres of $\pi$ are the tangent spaces $T_x(M)$ for $x\in M$.