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

• [ilmath]U[/ilmath] is saturated with respect to [ilmath]f[/ilmath]

if and only if

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

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