Separable

Separable is a mathematical term used to describe several related properties in topology and algebra. The exact meaning depends on the context, but all uses share a common theme of a notion of distinctness or density.

In topology, a space X is separable if it contains a countable dense subset. A subset D is dense if every nonempty open set in X meets D. The real numbers with the usual topology are separable because the rational numbers form a countable dense subset. Separable metric spaces have convenient properties: every separable metric space is second countable, which makes many arguments in analysis more manageable.

In algebra, separability applies to polynomials and field extensions. A polynomial over a field is separable if it has distinct roots in an algebraic closure; equivalently, the polynomial has no repeated roots, which can be detected by gcd(P, P′) = 1 in characteristic not equal to 0, and a corresponding derivative criterion holds in general. An algebraic extension E/F is separable if every element of E is a root of a separable polynomial over F, equivalently the minimal polynomial of each element over F has distinct roots. Finite separable extensions are characterized by the number of F-embeddings of E into an algebraic closure equaling the degree [E:F]. In characteristic p > 0, inseparable extensions can occur when minimal polynomials have repeated roots.

In ring theory and algebraic geometry, the notion extends to separable algebras. A finite type algebra A over a field k is separable if A is a separable k-algebra; over a field this often means A is a finite product of separable field extensions. Equivalently, A is projective as a module over A ⊗k A^op. This generalizes the concept of separable field extensions and aligns with étale morphisms in algebraic geometry.