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

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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[1]:

if and only if

  • Any one (or more) of the following:
    1. [ilmath]U=q^{-1}(q(U))[/ilmath][1]
    2. [ilmath]U[/ilmath] is a union of fibres[1]
    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][1]


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  1. 1.0 1.1 1.2 1.3 Introduction to Topological Manifolds - John M. Lee