Kernel (set theory)

{{short description|Equivalence relation expressing that two elements have the same image under a function}} {{other uses|Kernel (disambiguation)}}

In set theory, the '''kernel''' of a function <math>f</math> (or '''equivalence kernel'''<ref name="mac-lane-birkhoff">{{citation|title=Algebra|year=1999|first1=Saunders|last1=Mac Lane|author1link = Saunders Mac Lane|author2link = Garrett Birkhoff|first2=Garrett|last2=Birkhoff|publisher=Chelsea Publishing Company|isbn=0821816462|url=https://books.google.com/books?id=L6FENd8GHIUC&pg=PA33|pages=33}}.</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">{{citation|title=Universal Algebra: Fundamentals and Selected Topics|series=Pure and Applied Mathematics|volume=301|first=Clifford|last=Bergman|publisher=CRC Press|year=2011|isbn=9781439851296|url=https://books.google.com/books?id=QXi3BZWoMRwC&pg=PA14|pages=14–16}}.</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.

==Definition==

'''{{visible anchor|Kernel of a function}}'''

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 and only 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"/>

'''{{visible anchor|Kernel of a family of sets}}'''

The {{visible anchor|kernel of a family of sets|text='''kernel''' of a family <math>\mathcal{B} \neq \varnothing</math> of sets}} is{{sfn|Dolecki|Mynard|2016|pp=27–29, 33–35}} <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 {{em|{{visible anchor|fixed}}}} and is said to have {{em|{{visible anchor|non-empty intersection}}}} if its {{em|kernel}} is not empty.{{sfn|Dolecki|Mynard|2016|pp=27–29, 33–35}} A family is said to be {{em|{{visible anchor|free}}}} if it is not fixed; that is, if its kernel is the empty set.{{sfn|Dolecki|Mynard|2016|pp=27–29, 33–35}}

==Quotients==

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 product==

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 structures== {{See also|Kernel (algebra)}}

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 topology==

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">{{cite book|first=James|last=Munkres|authorlink = James Munkres|title=Topology|isbn=978-81-203-2046-8|publisher=Prentice-Hall of India|location=New Delhi|date=2004|page=169}}</ref><ref>{{planetmath| urlname=ASpaceIsCompactIffAnyFamilyOfClosedSetsHavingFipHasNonemptyIntersection|title=A space is compact iff any family of closed sets having fip has non-empty intersection}}</ref> said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.

==See also==

* {{annotated link|Filter on a set}}

==References==

==Bibliography==

* {{cite book|last=Awodey|first=Steve|authorlink=Steve Awodey|title=Category Theory|edition=2nd|orig-year=2006|year=2010|publisher=Oxford University Press|isbn=978-0-19-923718-0|series=Oxford Logic Guides|volume=49}} * {{Dolecki Mynard Convergence Foundations Of Topology}}

Category:Abstract algebra Category:Basic concepts in set theory Category:Set theory Category:Topology