Kernel (set theory)

In set theory, the kernel of a function $f$ (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 $f$ 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 ${\mathcal {B}},$ which by definition is the intersection of all its elements: $\ker {\mathcal {B}}~=~\bigcap _{B\in {\mathcal {B}}}\,B.$ 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 $f:X\to Y$ 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.^[2]

Kernel of a family of sets

The kernel of a family ${\mathcal {B}}\neq \varnothing$ of sets is^[3] $\ker {\mathcal {B}}~:=~\bigcap _{B\in {\mathcal {B}}}B.$ The kernel of ${\mathcal {B}}$ is also sometimes denoted by $\cap {\mathcal {B}}.$ 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.^[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: $\left\{\,\{w\in X:f(x)=f(w)\}~:~x\in X\,\right\}~=~\left\{f^{-1}(y)~:~y\in f(X)\right\}.$

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

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

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

Algebraic structures

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 $X.$ ^[2] 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

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;^[4]^[5] said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.

References

^ Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra, Chelsea Publishing Company, p. 33, ISBN 0821816462.
^ ^a ^b ^c ^d Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, vol. 301, CRC Press, pp. 14–16, ISBN 9781439851296.
^ ^a ^b ^c Dolecki & Mynard 2016, pp. 27–29, 33–35.
^ Munkres, James (2004). Topology. New Delhi: Prentice-Hall of India. p. 169. ISBN 978-81-203-2046-8.
^ 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.

[mac-lane-birkhoff-1] Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra, Chelsea Publishing Company, p. 33, ISBN 0821816462.

[bergman-2] Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, vol. 301, CRC Press, pp. 14–16, ISBN 9781439851296.

[FOOTNOTEDoleckiMynard201627–29,_33–35-3] Dolecki & Mynard 2016, pp. 27–29, 33–35.

[munkres-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.

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