In set theory, the kernel of a function (or equivalence kernel) may be taken to be either

• the equivalence relation on the function's domain 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 This definition is used in the theory of filters to classify them as being free or principal.

Definition

be a function between two sets are equivalent are equal is the equivalence relation thus defined.

The is 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

is called the coimage (or a variation). The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image specifically, the equivalence class).

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 (or a variation) and may be defined symbolically as

Algebraic structures

are algebraic structures of some fixed type (such as groupss, ringss, or vector spaces is a homomorphism is a congruence relation (that is an equivalence relation is a quotient is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.

In topology

is a continuous function between two topological spaces is a Hausdorff space must be a closed set 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 subsetss 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

References

Bibliography