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 $$\mathcal{B},$$ 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


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 $$\ker \mathcal{B} ~:=~ \bigcap_{B \in \mathcal{B}} B.$$ 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 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: $$\left\{\, \{w \in X : f(x) = f(w)\} ~:~ x \in X \,\right\} ~=~ \left\{f^{-1}(y) ~:~ y \in f(X)\right\}.$$

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 $$\ker f := \{(x,x') : f(x) = f(x')\}.$$

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 $$X.$$ 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.