Saturated set with respect to a function

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Definition

Let [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] are sets and let [ilmath]f:X\rightarrow Y[/ilmath] be any function between them. A subset of [ilmath]X[/ilmath], [ilmath]U\in\mathcal{P}(X)[/ilmath], is said to be saturated with respect to [ilmath]f[/ilmath] if:

• [ilmath]\exists V\in\mathcal{P}(Y)[U=f^{-1}(V)][/ilmath], in words:
  • There exists a subset of [ilmath]Y[/ilmath], [ilmath]V[/ilmath], such that [ilmath]V[/ilmath] is exactly the pre-image of [ilmath]U[/ilmath] under [ilmath]f[/ilmath]

See next

• Equivalent conditions to a set being saturated with respect to a map

See also

• Fibre - this (saturated set) is a generalisation of a fibre.