Kernel (set theory)

Short description: Equivalence relation expressing that two elements have the same image under a function

In set theory, the kernel of a function [math]\displaystyle{ f }[/math] (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 [math]\displaystyle{ f }[/math] can tell", 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.

Kernel of a family of sets

The kernel of a family [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] of sets is [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. 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: [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 [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] 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; 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

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.