In set theory, the kernel of a function f (or equivalence kernel ) may be taken to be either An unrelated notion is that of the kernel of a non-empty family of sets which by definition is the intersection of all its elements: This definition is used in the theory of filters to classify them as being free or principal.

Definition

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For the formal definition, let f : X \to Y be a function between two sets. Elements are equivalent if and are equal, that is, are the same element of Y. The kernel of f is the equivalence relation thus defined.

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The is The kernel of \mathcal{B} is also sometimes denoted by The kernel of the empty set, is typically left undefined. A family is called and is said to have if its is not empty. A family is said to be if it is not fixed; that is, if its kernel is the empty set.

Quotients

Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition: This quotient set X /=_f is called the coimage of the function f, and denoted (or a variation). The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, specifically, the equivalence class of x in X (which is an element of ) corresponds to f(x) in Y (which is an element of ).

As a subset of the Cartesian product

Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product X \times X. In this guise, the kernel may be denoted \ker f (or a variation) and may be defined symbolically as The study of the properties of this subset can shed light on f.

Algebraic structures

If X and Y are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function f : X \to Y is a homomorphism, then \ker f is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of f is a quotient of X. The bijection between the coimage and the image of f is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.

In topology

If f : X \to Y is a continuous function between two topological spaces then the topological properties of \ker f can shed light on the spaces X and Y. For example, if Y is a Hausdorff space then \ker f must be a closed set. Conversely, if X is a Hausdorff space and \ker f is a closed set, then the coimage of f, 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; said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.