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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} ".
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.
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 ...
An embedding h: N → M is called an elementary embedding of N into M if h(N) is an elementary substructure of M. A substructure N of M is elementary if and only if it passes the Tarski–Vaught test : every first-order formula φ ( x , b 1 , …, b n ) with parameters in N that has a solution in M also has a solution in N when evaluated in M .
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 ...
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 .
Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. [9] The following is a partial list of them: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. [10] For example, the union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.
In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a ...