Hi, given a function $f:X \rightarrow Y$, not necessarily invertible, is there a conventional name for the function $$g_f := f^{1} \circ f:X \rightarrow \mathcal{P}(X),$$ where $\mathcal{P}(X)$ denotes the power set of $X$?
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I do not know any name for it. However, note that the concept you are definining does not really depend on the function $f$, only on the equivalence relation induced by $f$ (which is sometimes called the kernel of $f$, unless you work with groups). A set $S$ is called "saturated" with respect to an equivalence relation $\theta$ iff $S$ is a union of equivalence classes, or equivalently, if $S=f^{1}(f(S))$ (where $f$ is some map inducing $\theta$). Hence "saturation" (as Benjamin Dickman suggested in a comment) would be a natural choice. But if the audience members are not mathematicians, "saturation" might mean something entirely different to them. I would suggest "$f$neighborhood"; this term can be easily visualized, I think. 

