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Kernel (set theory)
Equivalence relation expressing that two elements have the same image under a function

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

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

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Definition

Kernel of a function

For the formal definition, let f : X → Y {\displaystyle f:X\to Y} be a function between two sets. Elements x 1 , x 2 ∈ X {\displaystyle x_{1},x_{2}\in X} are equivalent if f ( x 1 ) {\displaystyle f\left(x_{1}\right)} and f ( x 2 ) {\displaystyle f\left(x_{2}\right)} are equal, that is, are the same element of Y . {\displaystyle Y.} The kernel of f {\displaystyle f} is the equivalence relation thus defined.3

Kernel of a family of sets

The kernel of a family B ≠ ∅ {\displaystyle {\mathcal {B}}\neq \varnothing } of sets is4 ker ⁡ B   :=   ⋂ B ∈ B B . {\displaystyle \ker {\mathcal {B}}~:=~\bigcap _{B\in {\mathcal {B}}}B.} The kernel of B {\displaystyle {\mathcal {B}}} is also sometimes denoted by ∩ B . {\displaystyle \cap {\mathcal {B}}.} The kernel of the empty set, ker ⁡ ∅ , {\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.5 A family is said to be free if it is not fixed; that is, if its kernel is the empty set.6

Quotients

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

This quotient set X / = f {\displaystyle X/=_{f}} is called the coimage of the function f , {\displaystyle f,} and denoted 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, im ⁡ f ; {\displaystyle \operatorname {im} f;} specifically, the equivalence class of x {\displaystyle x} in X {\displaystyle X} (which is an element of coim ⁡ f {\displaystyle \operatorname {coim} f} ) corresponds to f ( x ) {\displaystyle f(x)} in Y {\displaystyle Y} (which is an element of im ⁡ f {\displaystyle \operatorname {im} f} ).

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 X × X . {\displaystyle X\times X.} In this guise, the kernel may be denoted ker ⁡ f {\displaystyle \ker f} (or a variation) and may be defined symbolically as7 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 f . {\displaystyle f.}

Algebraic structures

See also: Kernel (algebra)

If X {\displaystyle X} and Y {\displaystyle Y} are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function f : X → Y {\displaystyle f:X\to Y} is a homomorphism, then 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 f {\displaystyle f} is a quotient of X . {\displaystyle X.} 8 The bijection between the coimage and the image of f {\displaystyle f} is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.

In topology

See also: Filters in topology

If f : X → Y {\displaystyle f:X\to Y} is a continuous function between two topological spaces then the topological properties of ker ⁡ f {\displaystyle \ker f} can shed light on the spaces X {\displaystyle X} and Y . {\displaystyle Y.} For example, if Y {\displaystyle Y} is a Hausdorff space then ker ⁡ f {\displaystyle \ker f} must be a closed set. Conversely, if X {\displaystyle X} is a Hausdorff space and ker ⁡ f {\displaystyle \ker f} is a closed set, then the coimage of 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;910 said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.

See also

Bibliography

  • 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.

References

  1. Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra, Chelsea Publishing Company, p. 33, ISBN 0821816462. 0821816462 ↩

  2. Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, vol. 301, CRC Press, pp. 14–16, ISBN 9781439851296. 9781439851296 ↩

  3. Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, vol. 301, CRC Press, pp. 14–16, ISBN 9781439851296. 9781439851296 ↩

  4. Dolecki & Mynard 2016, pp. 27–29, 33–35. - Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. https://search.worldcat.org/oclc/945169917 ↩

  5. Dolecki & Mynard 2016, pp. 27–29, 33–35. - Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. https://search.worldcat.org/oclc/945169917 ↩

  6. Dolecki & Mynard 2016, pp. 27–29, 33–35. - Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. https://search.worldcat.org/oclc/945169917 ↩

  7. Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, vol. 301, CRC Press, pp. 14–16, ISBN 9781439851296. 9781439851296 ↩

  8. Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, vol. 301, CRC Press, pp. 14–16, ISBN 9781439851296. 9781439851296 ↩

  9. Munkres, James (2004). Topology. New Delhi: Prentice-Hall of India. p. 169. ISBN 978-81-203-2046-8. 978-81-203-2046-8 ↩

  10. A space is compact iff any family of closed sets having fip has non-empty intersection at PlanetMath. https://planetmath.org/ASpaceIsCompactIffAnyFamilyOfClosedSetsHavingFipHasNonemptyIntersection ↩