Separabel

Separabel, often rendered separable in English, is a mathematical term used in several areas to denote a property that makes a structure in some sense built from distinct, non-redundant pieces. The most common usages are in topology/analysis and in field theory, where separability captures different but related ideas of “countable density” or “no repeated roots.”

In topology, a topological space is called separable if it contains a countable dense subset. A subset D of a space X is dense if every open set in X intersects D, equivalently the closure of D equals X. The real numbers with the standard topology provide a classic example: the rational numbers are countable and dense in R, so R is separable. In metric spaces, separability implies second countability, and in these spaces the two concepts are closely linked; however, in general topological spaces separability does not imply second countability.

In field theory, separability describes how algebraic elements behave with respect to a base field. An algebraic extension E/F is called separable if every element of E is a root of a polynomial over F that has distinct roots in a splitting field. Equivalently, for every element, its minimal polynomial over F has no repeated roots; equivalently, the polynomial f and its formal derivative f' are coprime. Over fields of characteristic zero (or when the base field is perfect), all finite extensions are separable. In positive characteristic, inseparable extensions can occur; for example, a root of X^p − t over a field of characteristic p may have derivative zero, making the polynomial inseparable and the extension inseparable.

Separable concepts also extend to related areas, such as separable polynomials, separable closures, and, in algebraic geometry, separable morphisms, which reflect a generalized notion of having distinct, non-redundant geometric behavior. See also separable extension and separable space.