Kernel (set theory)


In set theory, the kernel of a function f may be taken to be either
For the formal definition, let X and Y be sets and let f be a function from X to Y.
Elements x1 and x2 of X are equivalent if f and f are equal, i.e. are the same element of Y.
The kernel of f is the equivalence relation thus defined.

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 /= is called the coimage of the function, and denoted .
The coimage is naturally isomorphic to the image, ; specifically, the equivalence class of in corresponds to in .

As a subset of the square

Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product X × X.
In this guise, the kernel may be denoted and may be defined symbolically as
The study of the properties of this subset can shed light on.

In algebraic structures

If X and Y are algebraic structures of some fixed type, and if the function f from X to Y is a homomorphism, then ker f is a congruence relation, 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. See also Kernel.

In topological spaces

If X and Y are topological spaces and f is a continuous function between them, 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.