- Source: Kernel (set theory)
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.
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.
Kernel of a family of sets
The kernel of a family
B
≠
∅
{\displaystyle {\mathcal {B}}\neq \varnothing }
of sets is
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.
A family is said to be free if it is not fixed; that is, if its kernel is the empty set.
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 as
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
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.}
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
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; said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.
See also
Filter (set theory) – Family of sets representing "large" sets
References
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.
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