In set theory, the kernel of a function
(or equivalence kernel[1]) 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.
Kernel of a function For the formal definition, let
are equal, that is, are the same element of
is the equivalence relation thus defined.
[2] Kernel of a family of sets The kernel of a family
The kernel of the empty set,
is typically left undefined.
A family is called fixed and is said to have non-empty intersection if its kernel is not empty.
[3] A family is said to be free if it is not fixed; that is, if its kernel is the empty set.
[3] Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition:
is called the coimage of the function
The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image,
specifically, the equivalence class of
Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product
In this guise, the kernel may be denoted
(or a variation) and may be defined symbolically as[2]
The study of the properties of this subset can shed light on
are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function
is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of
[2] The bijection between the coimage and the image of
is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.
is a continuous function between two topological spaces then the topological properties of
can shed light on the spaces
is a Hausdorff space then
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 and only if every family of closed subsets with F.I.P.