Kernel (set theory)
Template:Short description {{#invoke:other uses|otheruses}}
In set theory, the kernel of a function <math>f</math> (or equivalence kernel<ref name="mac-lane-birkhoff">Template:Citation.</ref>) 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>f</math> can tell",<ref name="bergman">Template:Citation.</ref> or
- the corresponding partition of the domain.
An unrelated notion is that of the kernel of a non-empty family of sets <math>\mathcal{B},</math> which by definition is the intersection of all its elements: <math display=block>\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.
DefinitionEdit
For the formal definition, let <math>f : X \to Y</math> be a function between two sets. Elements <math>x_1, x_2 \in X</math> are equivalent if <math>f\left(x_1\right)</math> and <math>f\left(x_2\right)</math> are equal, that is, are the same element of <math>Y.</math> The kernel of <math>f</math> is the equivalence relation thus defined.<ref name="bergman"/>
The Template:Visible anchor isTemplate:Sfn <math display=block>\ker \mathcal{B} ~:=~ \bigcap_{B \in \mathcal{B}} B.</math> The kernel of <math>\mathcal{B}</math> is also sometimes denoted by <math>\cap \mathcal{B}.</math> The kernel of the empty set, <math>\ker \varnothing,</math> is typically left undefined. A family is called Template:Em and is said to have Template:Em if its Template:Em is not empty.Template:Sfn A family is said to be Template:Em if it is not fixed; that is, if its kernel is the empty set.Template:Sfn
QuotientsEdit
Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition: <math display=block>\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>X /=_f</math> is called the coimage of the function <math>f,</math> and denoted <math>\operatorname{coim} f</math> (or a variation). The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, <math>\operatorname{im} f;</math> specifically, the equivalence class of <math>x</math> in <math>X</math> (which is an element of <math>\operatorname{coim} f</math>) corresponds to <math>f(x)</math> in <math>Y</math> (which is an element of <math>\operatorname{im} f</math>).
As a subset of the Cartesian productEdit
Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product <math>X \times X.</math> In this guise, the kernel may be denoted <math>\ker f</math> (or a variation) and may be defined symbolically as<ref name="bergman"/> <math display=block>\ker f := \{(x,x') : f(x) = f(x')\}.</math>
The study of the properties of this subset can shed light on <math>f.</math>
Algebraic structuresEdit
If <math>X</math> and <math>Y</math> are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function <math>f : X \to Y</math> is a homomorphism, then <math>\ker f</math> is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of <math>f</math> is a quotient of <math>X.</math><ref name="bergman"/> The bijection between the coimage and the image of <math>f</math> is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.
In topologyEdit
If <math>f : X \to Y</math> is a continuous function between two topological spaces then the topological properties of <math>\ker f</math> can shed light on the spaces <math>X</math> and <math>Y.</math> For example, if <math>Y</math> is a Hausdorff space then <math>\ker f</math> must be a closed set. Conversely, if <math>X</math> is a Hausdorff space and <math>\ker f</math> is a closed set, then the coimage of <math>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;<ref name="munkres">Template:Cite book</ref><ref>Template:Planetmath</ref> said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.