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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} ".
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The family consisting only of the empty set and the set , called the minimal or trivial σ-algebra over . The power set of X , {\displaystyle X,} called the discrete σ-algebra . The collection { ∅ , A , X ∖ A , X } {\displaystyle \{\varnothing ,A,X\setminus A,X\}} is a simple σ-algebra generated by the subset A . {\displaystyle A.}
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.
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 .
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 ...
If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the ...
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 ...