Kernel (set theory)

{{visible anchor|Kernel of a function}}

For the formal definition, let $f\; :\; X\; \backslash to\; Y$ be a function between two sets.

Elements $x\_1,\; x\_2\; \backslash in\; X$ are equivalent if $f\backslash left(x\_1\backslash right)$ and $f\backslash left(x\_2\backslash right)$ are equal, that is, are the same element of $Y.$

The kernel of $f$ is the equivalence relation thus defined.

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

The {{visible anchor|kernel of a family of sets|text=kernel of a family $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ of sets}} is{{sfn|Dolecki|Mynard|2016|pp=27–29, 33–35}}

$$\backslash ker\; \backslash mathcal\{B\}\; ~:=~\; \backslash bigcap\_\{B\; \backslash in\; \backslash mathcal\{B\}\}\; B.$$

The kernel of $\backslash mathcal\{B\}$ is also sometimes denoted by $\backslash cap\; \backslash mathcal\{B\}.$ The kernel of the empty set, $\backslash ker\; \backslash varnothing,$ 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}}

Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition:

$$\backslash left\backslash \{\backslash ,\; \backslash \{w\; \backslash in\; X\; :\; f(x)\; =\; f(w)\backslash \}\; ~:~\; x\; \backslash in\; X\; \backslash ,\backslash right\backslash \}\; ~=~\; \backslash left\backslash \{f^\{-1\}(y)\; ~:~\; y\; \backslash in\; f(X)\backslash right\backslash \}.$$

This quotient set $X\; /=\_f$ is called the coimage of the function $f,$ and denoted $\backslash operatorname\{coim\}\; f$ (or a variation).

The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, $\backslash operatorname\{im\}\; f;$ specifically, the equivalence class of $x$ in $X$ (which is an element of $\backslash operatorname\{coim\}\; f$) corresponds to $f(x)$ in $Y$ (which is an element of $\backslash operatorname\{im\}\; f$).

Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product $X\; \backslash times\; X.$

In this guise, the kernel may be denoted $\backslash ker\; f$ (or a variation) and may be defined symbolically as

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

The study of the properties of this subset can shed light on $f.$

{{See also|Kernel (algebra)}}

If $X$ and $Y$ are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function $f\; :\; X\; \backslash to\; Y$ is a homomorphism, then $\backslash ker\; f$ 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.

{{See also|Filters in topology}}

If $f\; :\; X\; \backslash to\; Y$ is a continuous function between two topological spaces then the topological properties of $\backslash ker\; f$ can shed light on the spaces $X$ and $Y.$

For example, if $Y$ is a Hausdorff space then $\backslash ker\; f$ must be a closed set.

Conversely, if $X$ is a Hausdorff space and $\backslash ker\; f$ 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;{{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}}{{planetmath| urlname=ASpaceIsCompactIffAnyFamilyOfClosedSetsHavingFipHasNonemptyIntersection|title=A space is compact iff any family of closed sets having fip has non-empty intersection}} said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.

• {{annotated link|Filter (set theory)}}

{{reflist}}

• {{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}}

{{DEFAULTSORT:Kernel (Set Theory)}}

Category:Abstract algebra

Category:Basic concepts in set theory

Category:Set theory

Category:Topology