Kernel (set theory) explained

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.

l{B}, which by definition is the intersection of all its elements:$$\backslash ker\; \backslash mathcal\; ~=~\; \backslash bigcap\_\; \backslash ,\; B.$$ This definition is used in the theory of filters to classify them as being free or principal.

Definition

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]

The is $$\backslash ker\; \backslash mathcal\; ~:=~\; \backslash bigcap\_\; B.$$The kernel of

l{B} is also sometimes denoted by \cap l{B}. The kernel of the empty set, \ker \varnothing, is typically left undefined. A family is called and is said to have if its is not empty. A family is said to be 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:$$\backslash left\backslash \; ~=~\; \backslash left\backslash .$$

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

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

The study of the properties of this subset can shed light on

f.

Algebraic structures

See also: Kernel (algebra).

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

See also: Filters 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;^[3] said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.

Bibliography

• Book: Awodey, Steve. Steve Awodey

. Steve Awodey. Category Theory. 2nd. 2006. 2010. Oxford University Press. 978-0-19-923718-0. Oxford Logic Guides. 49.

Notes and References

1. .
2. .
3. Book: Munkres, James. Topology. 978-81-203-2046-8. Prentice-Hall of India. New Delhi. 2004. 169.