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Kernel (set theory)

## Summary

In set theory, the kernel of a function ${\displaystyle f}$ (or equivalence kernel[1]) 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 ${\displaystyle f}$ can tell",[2] or
• the corresponding partition of the domain.

An unrelated notion is that of the kernel of a non-empty family of sets ${\displaystyle {\mathcal {B}},}$ which by definition is the intersection of all its elements:

${\displaystyle \ker {\mathcal {B}}~=~\bigcap _{B\in {\mathcal {B}}}\,B.}$
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 ${\displaystyle f:X\to Y}$  be a function between two sets. Elements ${\displaystyle x_{1},x_{2}\in X}$  are equivalent if ${\displaystyle f\left(x_{1}\right)}$  and ${\displaystyle f\left(x_{2}\right)}$  are equal, that is, are the same element of ${\displaystyle Y.}$  The kernel of ${\displaystyle f}$  is the equivalence relation thus defined.[2]

Kernel of a family of sets

The kernel of a family ${\displaystyle {\mathcal {B}}\neq \varnothing }$  of sets is[3]

${\displaystyle \ker {\mathcal {B}}~:=~\bigcap _{B\in {\mathcal {B}}}B.}$

The kernel of ${\displaystyle {\mathcal {B}}}$  is also sometimes denoted by ${\displaystyle \cap {\mathcal {B}}.}$  The kernel of the empty set, ${\displaystyle \ker \varnothing ,}$  is typically left undefined. A family is called fixed and is said to have non-empty intersection if its kernel is not empty.[3] A family is said to be free if it is not fixed; that is, if its kernel is the empty set.[3]

## Quotients

Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition:

${\displaystyle \left\{\,\{w\in X:f(x)=f(w)\}~:~x\in X\,\right\}~=~\left\{f^{-1}(y)~:~y\in f(X)\right\}.}$

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

## 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 ${\displaystyle X\times X.}$  In this guise, the kernel may be denoted ${\displaystyle \ker f}$  (or a variation) and may be defined symbolically as[2]

${\displaystyle \ker f:=\{(x,x'):f(x)=f(x')\}.}$

The study of the properties of this subset can shed light on ${\displaystyle f.}$

## Algebraic structures

If ${\displaystyle X}$  and ${\displaystyle Y}$  are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function ${\displaystyle f:X\to Y}$  is a homomorphism, then ${\displaystyle \ker f}$  is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of ${\displaystyle f}$  is a quotient of ${\displaystyle X.}$ [2] The bijection between the coimage and the image of ${\displaystyle f}$  is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.

## In topology

If ${\displaystyle f:X\to Y}$  is a continuous function between two topological spaces then the topological properties of ${\displaystyle \ker f}$  can shed light on the spaces ${\displaystyle X}$  and ${\displaystyle Y.}$  For example, if ${\displaystyle Y}$  is a Hausdorff space then ${\displaystyle \ker f}$  must be a closed set. Conversely, if ${\displaystyle X}$  is a Hausdorff space and ${\displaystyle \ker f}$  is a closed set, then the coimage of ${\displaystyle f,}$  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;[4][5] said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.