In set theory, the kernel of a function <math>f</math> (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 <math>f</math> can tell", or
  • the corresponding partition of the domain.

An unrelated notion is that of the kernel of a non-empty family of sets <math>\mathcal{B},</math> which by definition is the intersection of all its elements:

<math display=block>\ker \mathcal{B} ~=~ \bigcap_{B \in \mathcal{B \, B.</math>

This definition is used in the theory of filters to classify them as being free or principal.

Definition

For the formal definition, let <math>f : X \to Y</math> be a function between two sets.

Elements <math>x_1, x_2 \in X</math> are equivalent if and only if <math>f\left(x_1\right)</math> and <math>f\left(x_2\right)</math> are equal, that is, are the same element of <math>Y.</math>

The kernel of <math>f</math> is the equivalence relation thus defined. 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

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