View source for Kernel (set theory)

{{short description|Equivalence relation expressing that two elements have the same image under a function}}


In [[Set theory|set theory]], the '''kernel''' of a [[Function (mathematics)|function]] <math>f</math> (or '''equivalence kernel'''<ref name="mac-lane-birkhoff">{{citation|title=Algebra|year=1999|first1=Saunders|last1=Mac Lane|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|equivalence relation]] on the function's [[Domain of a function|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 a set|partition]] of the domain.

An unrelated notion is that of the '''kernel''' of a non-empty [[Family of sets|family of sets]] <math>\mathcal{B},</math> which by definition is the [[Intersection|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 [[Filter (set theory)|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 [[Set (mathematics)|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"/> 

'''{{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|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 [[Ideal (ring theory)|modded out]] to form a [[Quotient set|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|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|bijection]]) to the [[Image (mathematics)|image]], <math>\operatorname{im} f;</math> specifically, the [[Equivalence class|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 square==

Like any [[Binary relation|binary relation]], the kernel of a function may be thought of as a [[Subset|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==

If <math>X</math> and <math>Y</math> are [[Algebraic structure|algebraic structure]]s of some fixed type (such as [[Group (mathematics)|group]]s, rings, or [[Vector space|vector space]]s), and if the function <math>f : X \to Y</math> is a [[Homomorphism|homomorphism]], then <math>\ker f</math> is a [[Congruence relation|congruence relation]] (that is an [[Equivalence relation|equivalence relation]] that is compatible with the algebraic structure), and the coimage of <math>f</math> is a [[Quotient (universal algebra)|quotient]] of <math>X.</math><ref name="bergman"/>
The bijection between the coimage and the image of <math>f</math> is an [[Isomorphism|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|continuous function]] between two [[Topological space|topological space]]s 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|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)|quotient space]] topology, must also be a Hausdorff space. 

A [[Topological space|space]] is [[Compact space|compact]] if and only if the kernel of every family of [[Closed set|closed subset]]s having the [[Finite intersection property|finite intersection property]] (FIP) is non-empty;<ref name="munkres">{{cite book|first=James|last=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 (set theory)}}

==References==

{{reflist}}

==Bibliography==

* {{cite book|last=Awodey|first=Steve|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}} <!--{{sfn|Dolecki|Mynard|2016|p=}}-->

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

{{Sourceattribution|Kernel (set theory)}}