Kernel (set theory)

For other uses, see Kernel (disambiguation).

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 Cartesian product

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

See also: Kernel (algebra)

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

See also: Filters 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.

See also

• Filter (set theory) – Family of sets representing "large" sets

References

1. Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra, Chelsea Publishing Company, p. 33, ISBN 0821816462.
2. Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, vol. 301, CRC Press, pp. 14–16, ISBN 9781439851296.
3. Dolecki & Mynard 2016, pp. 27–29, 33–35.
4. Munkres, James (2004). Topology. New Delhi: Prentice-Hall of India. p. 169. ISBN 978-81-203-2046-8.
5. A space is compact iff any family of closed sets having fip has non-empty intersection at PlanetMath.

Bibliography

• Awodey, Steve (2010) [2006]. Category Theory. Oxford Logic Guides. Vol. 49 (2nd ed.). Oxford University Press. ISBN 978-0-19-923718-0.
• Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.