Kernel (set theory)

In set theory, the kernel of a function "f" (or equivalence kernel) may be taken to be either
Definition.
For the formal definition, let "X" and "Y" be sets and let "f" be a function from "X" to "Y".
Elements "x"1 and "x"2 of "X" are "equivalent" if "f"("x"1) and "f"("x"2) are equal, i.e. are the same element of "Y".
The kernel of "f" is the equivalence relation thus defined.
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 (or a variation).
The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, ; specifically, the equivalence class of in (which is an element of ) corresponds to in (which is an element of ).
As a subset of the square.
Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product "X" × "X".
In this guise, the kernel may be denoted (or a variation) and may be defined symbolically as
The study of the properties of this subset can shed light on .
In algebraic structures.
If "X" and "Y" are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function "f" from "X" to "Y" is a homomorphism, then ker "f" is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of "f" is a quotient of "X".
The bijection between the coimage and the image of "f" is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem. See also Kernel (algebra).
In topological spaces.
If "X" and "Y" are topological spaces and "f" is a continuous function between them, then the topological properties of ker "f" can shed light on the spaces "X" and "Y".
For example, if "Y" is a Hausdorff space, then ker "f" must be a closed set.
Conversely, if "X" is a Hausdorff space and ker "f" is a closed set, then the coimage of "f", if given the quotient space topology, must also be a Hausdorff space.