Search results
Results From The WOW.Com Content Network
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} ".
If in the open set definition of "continuous map" (which is the statement: "every preimage of an open set is open"), both instances of the word "open" are replaced with "closed" then the statement of results ("every preimage of a closed set is closed") is equivalent to continuity.
This is also equivalent to any of {}, {<}, {} being measurable for all , or the preimage of any open set being measurable. Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable. [ 2 ]
In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map.
Let : be any function. If is any set then its preimage := under is necessarily an -saturated set.In particular, every fiber of a map is an -saturated set.. The empty set = and the domain = are always saturated.
If and are the domain and image of , respectively, then the fibers of are the sets in {():} = {{: =}:}which is a partition of the domain set .Note that must be restricted to the image set of , since otherwise () would be the empty set which is not allowed in a partition.
If e B is the neutral element of B, then the kernel of f is the preimage of the singleton set {e B}; that is, the subset of A consisting of all those elements of A that are mapped by f to the element e B. The kernel is usually denoted ker f (or a variation). In symbols: = {: =}.
An ordered pair (,), where is a set and is a σ-algebra over , is called a measurable space. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable.