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The preimage of an output value is the set of input values that produce . More generally, evaluating f {\displaystyle f} at each element of a given subset A {\displaystyle A} of its domain X {\displaystyle X} produces a set, called the " image of A {\displaystyle A} under (or through) f {\displaystyle f} ".
The fibers of are that line and all the straight lines parallel to it, which form a partition of the plane . More generally, if f {\displaystyle f} is a linear map from some linear vector space X {\displaystyle X} to some other linear space Y {\displaystyle Y} , the fibers of f {\displaystyle f} are affine subspaces of X {\displaystyle X ...
Then a pullback of f and g (in Set) is given by the preimage f −1 [B 0] together with the inclusion of the preimage in A. f −1 [B 0] ↪ A. and the restriction of f to f −1 [B 0] f −1 [B 0] → B 0. Because of this example, in a general category the pullback of a morphism f and a monomorphism g can be thought of as the "preimage" under ...
Moreover, f is the composition of the canonical projection from f to the quotient set, and the bijection between the quotient set and the codomain of . The composition of two surjections is again a surjection, but if g ∘ f {\displaystyle g\circ f} is surjective, then it can only be concluded that g {\displaystyle g} is surjective (see figure).
If f : X → Y is surjective and B is a subset of Y, then f(f −1 (B)) = B. Thus, B can be recovered from its preimage f −1 (B). For example, in the first illustration in the gallery, there is some function g such that g(C) = 4. There is also some function f such that f(4) = C.
Some authors call a function : between two topological spaces proper if the preimage of every compact set in is compact in . Other authors call a map f {\displaystyle f} proper if it is continuous and closed with compact fibers ; that is if it is a continuous closed map and the preimage of every point in Y {\displaystyle Y} is compact .
After a Form 3 is filed, future filings of the same nature are filed under Form 4 (standard disclosure) or Form 5 (annual disclosure). Form 3 is stored in SEC's EDGAR database and academic researchers make these reports freely available as structured datasets in the Harvard Dataverse. [1] [2] [3]
Representations of the fundamental group have a very geometric significance: any local system (i.e., a sheaf on X with the property that locally in a sufficiently small neighborhood U of any point on X, the restriction of F is a constant sheaf of the form | =) gives rise to the so-called monodromy representation, a representation of the ...