Kernel (set theory)

In set theory, the kernel of a function 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 can tell", or
the corresponding partition of the domain.

An unrelated notion is that of the kernel of a non-empty family of sets which by definition is the intersection of all its elements:
This definition is used in the theory of filters to classify them as being free or principal.

Definition

'
For the formal definition, let be a function between two sets.
Elements are equivalent if and are equal, that is, are the same element of
The kernel of is the equivalence relation thus defined.
'
The is
The kernel of is also sometimes denoted by The kernel of the empty set, is typically left undefined.
A family is called and is said to have if its is not empty.
A family is said to be 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:
This quotient set is called the coimage of the function and denoted .
The coimage is naturally isomorphic to the image, specifically, the equivalence class of in corresponds to in .

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
In this guise, the kernel may be denoted and may be defined symbolically as
The study of the properties of this subset can shed light on

Algebraic structures

If and are algebraic structures of some fixed type, and if the function is a homomorphism, then is a congruence relation, and the coimage of is a quotient of
The bijection between the coimage and the image of is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.

In topology

If is a continuous function between two topological spaces then the topological properties of can shed light on the spaces and
For example, if is a Hausdorff space then must be a closed set.
Conversely, if is a Hausdorff space and is a closed set, then the coimage of if given the space (topology)|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 is non-empty; said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.