Separable

Let f(x) be an irreducible polynomial with coefficients in a field K. Then f(x) is said to be separable over K if it has no repeated zeros in a splitting field. A general polynomial f(x) in K[x] is called separable iff all its irreducible factors are.

For example, if we work over any field extension of Q (i.e. a field of characteristic zero) then all polynomials are separable. But field extensions of finite fields can have inseparable polynomials. The first of these claims follows easily from

Lemma Let f be a nonzero polynomial over some field k. Then f has a repeated zero in a splitting field of f over k iff f and df/dx have a common factor of degree > 0 in k[x].

The lemma is easily proved using the formula for differentiating a product. Bear in mind though that funny things can happen in characteristic p. For example x^p-1 in Z_p[x] differentiates to zero!

Finally, an algebraic field extension M of K is called separable if the minimal polynomial over K of each element of M is separable over K.

Any algebraic extension of a subfield of C is separable.

Mathematicians hate to let a good word go to waste, and so separability is also an axiom in topology.

Definition A topological space X is said to be separable if X has a countable dense subset. In other words, there is a countable subset D of X such that closure(D) = X. Equivalently, each nonempty open set in X intersects D.

Recall that a space is second countable if there is a countable basis for the space.

Theorem Let X be a topological space. If X is second countable then X is separable. If X is separable and metrizable, then X is second countable (and hence Lindelof).

A countable product of separable spaces is separable, but a subspace of a separable space is not necessarily separable; a counterexample can be found in the Sorgenfrey plane. However, a metrizable separable space is second countable, and so any subspace is second countable, and therefore separable.

We also have:

Theorem A compact metrizable space is separable.

One use of separability is this: suppose X,Y are topological spaces, f:X->Y is continuous and D is dense in X. Then f is uniquely determined by it's behaviour on D. If D is countable, then this lets us restrict our attention to a countable subset of X, which is generally much easier to handle than the whole space, and allows use of induction.

Some examples of separable spaces in topology and analysis:

• A countable space is separable
• Euclidean space Rⁿ in the usual topology is separable, with countable dense subset Qⁿ
• The space beta(N), the Stone-Cech compactification of the natural numbers, is separable
• The Banach space C([0,1]) of continuous functions on [0,1] is separable, with countable dense subset the set of polynomials on [0,1] with rational coefficients (this is the Weierstrass Approximation Theorem)
• The Hilbert space L²([0,2pi]) of square integrable functions f:[0,2pi]->C with the 2-norm is separable. This is of interest in analysis, since it can be shown that any two separable Hilbert spaces are isometrically isomorphic. See Fourier series

Spaces that aren't separable include:

• The first uncountable ordinal omega₁ in the order topology
• The space X^J in the product (or box) topology, where X is a topological space with at least two points and J is uncountable
• The Banach space l^infinity of bounded complex sequences with the supremum norm

For a finite dimensional normed space, many of the computations can be simplified by the existence of a finite basis, where every vector in the space is a linear combination of some subset of vectors in the basis. It is possible to extend this idea of a basis to infinite dimensional normed spaces in certain situations. Many infinite dimensional spaces contain an infinite sequence of vectors, known as a fundamental sequence, such that any vector in that space can be approximated by linear combinations of vectors belonging to that fundamental sequence. A normed space is called separable if a fundamental sequence exists.

Many infinite dimensional spaces are easily seen to be separable. For instance, the normed space of square integrable functions with the L₂ norm on the unit circle in the complex plane is separable, and its fundamental sequence consists of complex exponential functions, or, the Fourier series of the function. This is shown by the Riesz-Fischer Theorem and forms the basis for Fourier analysis.

The space of all infinitely differentiable functions is also separable, and a fundamental sequence is the sequence of all powers of x. This is shown by Taylor series and Weierstrass's theorem.

Other separable spaces include the l² space of absolute square summable infinite sequences of complex numbers is also obviously separable. The fundamental sequence is clearly the sequence of vectors whose nth number is 1, and all other numbers are zero. The spaces used in wavelet theory are also separable.

One example of a non-separable space is the l^∞ space which is the same as l² except that the norm is ||(a₀, a₁, a₂...)||_∞ = sup a_n. This space can be shown to be not separable. The proof is not trivial though.

Let's talk about the topological definition given by Evandar a bit more. Topological spaces are rather abstract, but the ones that we're mostly interested in are metrisable, which means they can be made into a metric space. You should think of a metric space M as a set of points together with some sort of way of measuring the "distance" between any 2 of them. This "distance" is just a function from MxM to the positve real numbers that satisfies certain simple axioms we want the distance to satisfy. The real number line, with the distance between 2 points x and y given by |x-y|, is a typical example.

 

Now "countable" means "small" in mathematics, for various reasons. The real numbers are not countable, but some subsets of them are, for example the integers or rationals. It's the rationals we want to think about here, because they are also what's called dense. For real numbers, this just means that in any interval you can always find one. In a general metric space, a set is dense if every point in the whole space can be written as the limit of a sequence of points, each element lying in the dense set in question. So every real is the limit of a sequence of rationals (just truncate the decimal expansion). It's not too hard to see this is equivalent to the first thing I said.

 

 

Dense means "large" in some sense, because every point in the whole space is as close as you like to a point in the dense set. So if a set is countable AND dense, it's both "large" and "small". The interplay between different notions of the "size" of a set is fundamental to lots of analysis.

 

Not all metric spaces are separable, for example the set of real numbers with the "discrete metric". So if a space does happen to be separable, it's really useful, because we only need understand a "small" (countable) set to understand lots about the whole space. This allows us to prove theorems more easily. For example, these ideas lead to a proof that if f is a continuous function on a closed bounded interval and if the integral of (f(x) multiplied by xⁿ) over this interval is 0 for every n, then f must be identically zero. This is known as the moments problem.

Sep"a*ra*ble (?), a. [L. separabilis: cf. F. s'eparable.]

Capable of being separated, disjoined, disunited, or divided; as, the separable parts of plants; qualities not separable from the substance in which they exist.

-- Sep"a*ra*ble*ness, n. -- Sep"a*ra*bly, adv.

Trials permit me not to doubt of the separableness of a yellow tincture from gold. Boyle.

 

© Webster 1913.