# Kernel (set theory)

__: Equivalence relation expressing that two elements have the same image under a function__

**Short description**

In set theory, the **kernel** of a function [math]\displaystyle{ f }[/math] (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 [math]\displaystyle{ f }[/math] 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 [math]\displaystyle{ \mathcal{B}, }[/math] which by definition is the intersection of all its elements:
[math]\displaystyle{ \ker \mathcal{B} ~=~ \bigcap_{B \in \mathcal{B}} \, B. }[/math]
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 [math]\displaystyle{ f : X \to Y }[/math] be a function between two sets.
Elements [math]\displaystyle{ x_1, x_2 \in X }[/math] are *equivalent* if [math]\displaystyle{ f\left(x_1\right) }[/math] and [math]\displaystyle{ f\left(x_2\right) }[/math] are equal, that is, are the same element of [math]\displaystyle{ Y. }[/math]
The kernel of [math]\displaystyle{ f }[/math] is the equivalence relation thus defined.^{[2]}

**Kernel of a family of sets**

The **kernel** of a family [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] of sets is^{[3]}
[math]\displaystyle{ \ker \mathcal{B} ~:=~ \bigcap_{B \in \mathcal{B}} B. }[/math]
The kernel of [math]\displaystyle{ \mathcal{B} }[/math] is also sometimes denoted by [math]\displaystyle{ \cap \mathcal{B}. }[/math] The kernel of the empty set, [math]\displaystyle{ \ker \varnothing, }[/math] 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: [math]\displaystyle{ \left\{\, \{w \in X : f(x) = f(w)\} ~:~ x \in X \,\right\} ~=~ \left\{f^{-1}(y) ~:~ y \in f(X)\right\}. }[/math]

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

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

The study of the properties of this subset can shed light on [math]\displaystyle{ f. }[/math]

## Algebraic structures

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

## In topology

If [math]\displaystyle{ f : X \to Y }[/math] is a continuous function between two topological spaces then the topological properties of [math]\displaystyle{ \ker f }[/math] can shed light on the spaces [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y. }[/math] For example, if [math]\displaystyle{ Y }[/math] is a Hausdorff space then [math]\displaystyle{ \ker f }[/math] must be a closed set. Conversely, if [math]\displaystyle{ X }[/math] is a Hausdorff space and [math]\displaystyle{ \ker f }[/math] is a closed set, then the coimage of [math]\displaystyle{ f, }[/math] 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

- ↑ Mac Lane, Saunders; Birkhoff, Garrett (1999),
*Algebra*, Chelsea Publishing Company, pp. 33, ISBN 0821816462, https://books.google.com/books?id=L6FENd8GHIUC&pg=PA33. - ↑
^{2.0}^{2.1}^{2.2}^{2.3}Bergman, Clifford (2011),*Universal Algebra: Fundamentals and Selected Topics*, Pure and Applied Mathematics,**301**, CRC Press, pp. 14–16, ISBN 9781439851296, https://books.google.com/books?id=QXi3BZWoMRwC&pg=PA14. - ↑
^{3.0}^{3.1}^{3.2}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.org.

## Bibliography

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

Original source: https://en.wikipedia.org/wiki/Kernel (set theory).
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