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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} ".
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.
A Vitali set is a subset of the interval [,] of real numbers such that, for each real number , there is exactly one number such that is a rational number.Vitali sets exist because the rational numbers form a normal subgroup of the real numbers under addition, and this allows the construction of the additive quotient group / of these two groups which is the group formed by the cosets + of the ...
In words, given two programs, if the first program is in the set of programs satisfying the property and two programs are computing the same thing, then also the second program satisfies the property. This means that if one program with a certain property is in the set, all programs computing the same function must also be in the set).
Consider the set of real numbers with the ordinary topology, and write if and only if is an integer. Then the quotient space X / ∼ {\displaystyle X/{\sim }} is homeomorphic to the unit circle S 1 {\displaystyle S^{1}} via the homeomorphism which sends the equivalence class of x {\displaystyle x} to exp ( 2 π i x ) . {\displaystyle \exp(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.
In mathematics, more specifically in point-set topology, the derived set of a subset of a topological space is the set of all limit points of . It is usually denoted by S ′ . {\displaystyle S'.} The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line .
Conversely, if is a Hausdorff space and is a closed set, then the coimage of , if given the quotient space topology, must also be a Hausdorff space. A space is compact if and only if the kernel of every family of closed subsets having the finite intersection property (FIP) is non-empty; [ 4 ] [ 5 ] said differently, a space is compact if ...