An algebraic closure K < sup > alg </ sup > of K contains a unique separable extension K < sup > sep </ sup > of K containing all ( algebraic ) separable extensions of K within K < sup > alg </ sup >.

This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of K < sup > sep </ sup >, of degree > 1.

A first countable, separable Hausdorff space ( in particular, a separable metric space ) has at most the continuum cardinality c. In such a space, closure is determined by limits of sequences and any sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of X.

Given a separable extension K ′ of K, a Galois closure L of K ′ is a type of splitting field, and also a Galois extension of K containing K ′ that is minimal, in an obvious sense.

For example the algebraic closure, the separable closure, the cyclic closure et cetera.

According to the most common one, a polynomial P ( X ) over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to its degree.

For the second definition, P ( X ) is separable if all of its irreducible factors in K have distinct roots in the splitting field of P ( X ), or equivalently in an algebraic closure of K. For this definition, separability depends explicitly on the field K, as an irreducible polynomial P which is not separable becomes separable over the splitting field of K. Also, for this definition every polynomial over a perfect field is separable, which includes in particular all fields of characteristic 0, and all finite fields.

If F is any field, the separable closure F < sup > sep </ sup > of F is the field of all elements in an algebraic closure of F that are separable over F. This is the maximal Galois extension of F. By definition, F is perfect if and only if its separable and algebraic closures coincide ( in particular, the notion of a separable closure is only interesting for imperfect fields ).

If X ′ is separable, then X is separable.

Then X is separable if and only if X ′ is separable.

* Every compact metric space is separable.

Another active area of research is the program to obtain classification, or to determine the extent of which classification is possible, for separable simple nuclear C *- algebras.

A topological space homeomorphic to a separable complete metric space is called a Polish space.

While most Protestants agree that baptism in the Holy Spirit is integral to being a Christian, others believe that it is not separable from conversion and no longer marked by glossolalia.

Separable polynomials are used to define separable extensions: A field extension is a separable extension if and only if for every, which is algebraic over K, the minimal polynomial of over K is a separable polynomial.

In modern algebra, an algebraic field extension is a separable extension if and only if for every, the minimal polynomial of over F is a separable polynomial ( i. e., has distinct roots ).

There are other equivalent definitions of the notion of a separable algebraic extension, and these are outlined later in the article.

Since algebraic extensions of fields of characteristic zero, and of finite fields, are separable, separability is not an obstacle in most applications of Galois theory.

For instance, every algebraic ( in particular, finite degree ) extension of the field of rational numbers is necessarily separable.

An algebraic extension is a purely inseparable extension if and only if for every, the minimal polynomial of over F is not a separable polynomial ( i. e., does not have distinct roots ).

For a field F to possess a non-trivial purely inseparable extension, it must necessarily be an infinite field of prime characteristic ( i. e. specifically, imperfect ), since any algebraic extension of a perfect field is necessarily separable.

In fact, it is true that there is no irreducible polynomial with rational or real coefficients that does not have distinct roots ; in the language of field theory, every algebraic extension of or is separable and hence both of these fields are perfect.

A field F is perfect if and only if all of its algebraic extensions are separable ( in fact, all algebraic extensions of F are separable if and only if all finite degree extensions of F are separable ).

Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is regular, Hausdorff and second-countable.

This state is separable if yielding and It is inseparable if If a state is inseparable, it is called an entangled state.

Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as

In other words, a state is separable if it is probability distribution over uncorrelated states, or product states.

A state is then said to be entangled if it is not separable.

In mathematics a topological space is called separable if it contains a countable dense subset ; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

Any second-countable space is separable: if is a countable base, choosing any gives a countable dense subset.

Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf.

* A Hilbert space is separable if and only if it has a countable orthonormal basis, it follows that any separable, infinite-dimensional Hilbert space is isometric to ℓ < sup > 2 </ sup >.

A phrase that is often used to describe the relationship between the EU forces and NATO is " separable, but not separate ": The same forces and capabilities form the basis of both EU and NATO efforts, but portions can be allocated to the European Union if necessary.

The perceptron learning algorithm does not terminate if the learning set is not linearly separable.

Separability is especially important in numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces.

As a consequence of the Weierstrass approximation theorem, one can show that the space C is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients ; there are only countably many polynomials with rational coefficients.

Generally speaking, constructive analysis can reproduce theorems of classical analysis, but only in application to separable spaces ; also, some theorems may need to be approached by approximations.

A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable.

In order to avoid the number of histogram bins growing like the probability distribution is approximated by a separable function: so that the number of bins required is only Kd.

Many of the variations are separable only on tail and wing lengths and these vary with overlap across populations.

A polynomial P ( X ) is separable if and only if it is coprime to its formal derivative P ′( X ).

Inseparable extensions ( that is extensions which are not separable ) may occur only in characteristic p.

More specifically, a finite degree field extension is Galois if and only if it is both normal and separable.

A polynomial f in F is a separable polynomial if and only if every irreducible factor of f in F has distinct roots.