Kernel (set theory)

For other uses, see Kernel (disambiguation).

In set theory, the kernel of a function $f$ (or equivalence kernel) may be taken to be either

• the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function $f$ can tell", or
• the corresponding partition of the domain.

An unrelated notion is that of the kernel of a non-empty family of sets $ℬ,$ which by definition is the intersection of all its elements: $$\ker ℬ=\bigcap _{B\in ℬ}B.$$ This definition is used in the theory of filters to classify them as being free or principal.

Definition

Kernel of a function

For the formal definition, let $f:X\to Y$ be a function between two sets. Elements $x_1,x_2\in X$ are equivalent if $f\left(x_1\right)$ and $f\left(x_2\right)$ are equal, that is, are the same element of $Y.$ The kernel of $f$ is the equivalence relation thus defined.

Kernel of a family of sets

The kernel of a family $ℬ\ne \varnothing$ of sets is $$\ker ℬ:=\bigcap _{B\in ℬ}B.$$ The kernel of $ℬ$ is also sometimes denoted by $\cap ℬ.$ The kernel of the empty set, $\ker \varnothing ,$ is typically left undefined. A family is called fixed and is said to have non-empty intersection if its kernel is not empty. A family is said to be free 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: $$\{\{w\in X:f(x)=f(w)\}:x\in X\}=\{f^{-1}(y):y\in f(X)\}.$$

This quotient set $X/{=}_f$ is called the coimage of the function $f,$ and denoted $\mathrm{coim}f$ (or a variation). The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, $\mathrm{im}f;$ specifically, the equivalence class of $x$ in $X$ (which is an element of $\mathrm{coim}f$) corresponds to $f(x)$ in $Y$ (which is an element of $\mathrm{im}f$).

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 $$\ker f:=\{(x,x^{\prime }):f(x)=f(x^{\prime })\}.$$

The study of the properties of this subset can shed light on $f.$

Algebraic structures

See also: Kernel (algebra)

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

See also: Filters 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.

See also

• Filter on a set – Family of subsets representing "large" sets