Kernel (set theory)

In set theory, the kernel of a function ${\displaystyle f}${\displaystyle f} (or equivalence kernel^[1] ) 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 ${\displaystyle f}${\displaystyle f} can tell",^[2] or
• the corresponding partition of the domain.

An unrelated notion is that of the kernel of a non-empty family of sets ${\displaystyle {\mathcal {B}},}${\displaystyle {\mathcal {B}},} which by definition is the intersection of all its elements: $${\displaystyle \ker {\mathcal {B}}~=~\bigcap _{B\in {\mathcal {B}}},円B.}$${\displaystyle \ker {\mathcal {B}}~=~\bigcap _{B\in {\mathcal {B}}},円B.} This definition is used in the theory of filters to classify them as being free or principal.

Kernel of a function

For the formal definition, let ${\displaystyle f:X\to Y}${\displaystyle f:X\to Y} be a function between two sets. Elements ${\displaystyle x_{1},x_{2}\in X}${\displaystyle x_{1},x_{2}\in X} are equivalent if ${\displaystyle f\left(x_{1}\right)}${\displaystyle f\left(x_{1}\right)} and ${\displaystyle f\left(x_{2}\right)}${\displaystyle f\left(x_{2}\right)} are equal, that is, are the same element of ${\displaystyle Y.}${\displaystyle Y.} The kernel of ${\displaystyle f}${\displaystyle f} is the equivalence relation thus defined.^[2]

Kernel of a family of sets

The kernel of a family ${\displaystyle {\mathcal {B}}\neq \varnothing }${\displaystyle {\mathcal {B}}\neq \varnothing } of sets is^[3] $${\displaystyle \ker {\mathcal {B}}~:=~\bigcap _{B\in {\mathcal {B}}}B.}$${\displaystyle \ker {\mathcal {B}}~:=~\bigcap _{B\in {\mathcal {B}}}B.} The kernel of ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is also sometimes denoted by ${\displaystyle \cap {\mathcal {B}}.}${\displaystyle \cap {\mathcal {B}}.} The kernel of the empty set, ${\displaystyle \ker \varnothing ,}${\displaystyle \ker \varnothing ,} is typically left undefined. A family is called fixed and is said to have non-empty intersection if its kernel is not empty.^[3] A family is said to be free if it is not fixed; that is, if its kernel is the empty set.^[3]

Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition: $${\displaystyle \left\{,円\{w\in X:f(x)=f(w)\}~:~x\in X,円\right\}~=~\left\{f^{-1}(y)~:~y\in f(X)\right\}.}$${\displaystyle \left\{,円\{w\in X:f(x)=f(w)\}~:~x\in X,円\right\}~=~\left\{f^{-1}(y)~:~y\in f(X)\right\}.}

This quotient set ${\displaystyle X/=_{f}}${\displaystyle X/=_{f}} is called the coimage of the function ${\displaystyle f,}${\displaystyle f,} and denoted ${\displaystyle \operatorname {coim} f}${\displaystyle \operatorname {coim} f} (or a variation). The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, ${\displaystyle \operatorname {im} f;}${\displaystyle \operatorname {im} f;} specifically, the equivalence class of ${\displaystyle x}${\displaystyle x} in ${\displaystyle X}${\displaystyle X} (which is an element of ${\displaystyle \operatorname {coim} f}${\displaystyle \operatorname {coim} f}) corresponds to ${\displaystyle f(x)}${\displaystyle f(x)} in ${\displaystyle Y}${\displaystyle Y} (which is an element of ${\displaystyle \operatorname {im} f}${\displaystyle \operatorname {im} f}).

Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product ${\displaystyle X\times X.}${\displaystyle X\times X.} In this guise, the kernel may be denoted ${\displaystyle \ker f}${\displaystyle \ker f} (or a variation) and may be defined symbolically as^[2] $${\displaystyle \ker f:=\{(x,x'):f(x)=f(x')\}.}$${\displaystyle \ker f:=\{(x,x'):f(x)=f(x')\}.}

The study of the properties of this subset can shed light on ${\displaystyle f.}${\displaystyle f.}

If ${\displaystyle X}${\displaystyle X} and ${\displaystyle Y}${\displaystyle Y} are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function ${\displaystyle f:X\to Y}${\displaystyle f:X\to Y} is a homomorphism, then ${\displaystyle \ker f}${\displaystyle \ker f} is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of ${\displaystyle f}${\displaystyle f} is a quotient of ${\displaystyle X.}${\displaystyle X.}^[2] The bijection between the coimage and the image of ${\displaystyle f}${\displaystyle f} is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.

If ${\displaystyle f:X\to Y}${\displaystyle f:X\to Y} is a continuous function between two topological spaces then the topological properties of ${\displaystyle \ker f}${\displaystyle \ker f} can shed light on the spaces ${\displaystyle X}${\displaystyle X} and ${\displaystyle Y.}${\displaystyle Y.} For example, if ${\displaystyle Y}${\displaystyle Y} is a Hausdorff space then ${\displaystyle \ker f}${\displaystyle \ker f} must be a closed set. Conversely, if ${\displaystyle X}${\displaystyle X} is a Hausdorff space and ${\displaystyle \ker f}${\displaystyle \ker f} is a closed set, then the coimage of ${\displaystyle f,}${\displaystyle 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;^{[4] [5]} said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.

• Filter on a set – Family of subsets representing "large" sets

• Awodey, Steve (2010) [2006]. Category Theory. Oxford Logic Guides. Vol. 49 (2nd ed.). Oxford University Press. ISBN 978-0-19-923718-0.
• Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.

• Abstract algebra
• Basic concepts in set theory
• Set theory
• Topology

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