Equivalent conditions to a set being saturated with respect to a function

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Review and demote as necessary

Statement

Let $X$ and $Y$ be sets and let $f:X\rightarrow Y$ be a function. Let $U\in\mathcal{P}(X)$ be an arbitrary subset of $X$, then:

• $U$ is saturated with respect to $f$

if and only if

• Any one (or more) of the following:
  1. $U=q^{-1}(q(U))$
  2. $U$ is a union of fibres
  3. if $x\in U$ then every point $x'\in X$ such that $q(x)=q(x')$ is also in $U$

Proof

Grade: A

This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:

Easy work, routine

This proof has been marked as an page requiring an easy proof