Kernel (set theory)

In set theory, the kernel of a function $f$ (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 $f$ can tell",. or
* the corresponding partition of the domain.

An unrelated notion is that of the kernel of a non-empty family of sets $\mathcal,$ which by definition is the intersection of all its elements:
$\ker \mathcal ~=~ \bigcap_ \, B.$
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.

The is
$\ker \mathcal ~:=~ \bigcap_ B.$
The kernel of $\mathcal$ is also sometimes denoted by $\cap \mathcal.$ 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:
$\left\ ~=~ \left\.$

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

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
$\ker f := \.$

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 $X.$
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; said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.

See also

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References

Bibliography

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Abstract algebra
Basic concepts in set theory
Set theory
Topology