Changes

Let's say that we have an ordinary function f : X -> Y.  If we pick out one y from the target or the codomain Y, then it either has x's from the source or the domain X that are assigned, or mapped, or sent to it by f or it doesn't.  Let's say it does.  Then here is the picture of what I will frequently call the "fiber (of f) at y":

Let's say that we have an ordinary function f : X -> Y.  If we pick out one y from the target or the codomain Y, then it either has x's from the source or the domain X that are assigned, or mapped, or sent to it by f or it doesn't.  Let's say it does.  Then here is the picture of what I will frequently call the "fiber (of f) at y":

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Very often the reason that one is interested in these varieties of fibers under a given function is so that one can follow them "upstream" or "backward", functionally speaking, in other words, toward the "source" of the functional value under investigation.  That leads rather naturally to the other mathematical usage for the word "fiber" that I have in mind.  Here are the definitions as I formulated them in my dissertation proposal:

Very often the reason that one is interested in these varieties of fibers under a given function is so that one can follow them "upstream" or "backward", functionally speaking, in other words, toward the "source" of the functional value under investigation.  That leads rather naturally to the other mathematical usage for the word "fiber" that I have in mind.  Here are the definitions as I formulated them in my dissertation proposal: