Saturation and Definition of Fibers in Mathematics 📂Set Theory

Saturation and Definition of Fibers in Mathematics

Definitions

Given two sets $X$, $Y$ and a function $\pi\ :\ X\rightarrow Y$. If $\pi^{-1}\big( \pi (u) \big)=u$ holds, then $u\subset X$ is called saturation.
The set $\pi^{-1}(y) \subset X$ is called the fiber or stalk over the point $y\in Y$ in $\pi$.

$\pi^{-1}$ is a preimage. Let’s easily understand through the pictures below.

$u$ is always less than or equal to $\pi^{-1} \big( \pi (u) \big)$. Thus, being saturated means that $u$ has grown as much as it possibly can.

In simple terms, it’s a preimage for a point. The reason why it’s called a fiber is intuitively clear from the picture below.

Also, by the two definitions, it is easy to see the following fact.

$u \subset X$ being saturated is equivalent to $u$ being the union of the fibers of $\pi\ :\ X\rightarrow Y$.