In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. This condition is typical of spaces that are met in classical parts of mathematical analysis and geometry. In the same way that any real number can be approximated to any specified accuracy by rational numbers, a separable space has some countable subset with which all its elements can be approached, in the sense of a mathematical limit. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, the term dense has at least three different meanings. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ... Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ... The text or formatting below is generated by a template which has been proposed for deletion. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...

Separable spaces are topological spaces with a certain limitation on their size. The separability property is often listed as one of the axioms of countability. From an axiomatic point of view separability was rather frowned upon in the period 1940 to 1960 — where previously it had been basic to descriptive set theory. Subsequently the pendulum swung back, and textbooks would more often choose to admit separability, proving less general theorems (this attitude was adopted, for example, by Jean Dieudonné). For example taking Hilbert space to mean a complex Hilbert space of infinite dimension and separable, there is one such space up to isomorphism (there is a categorical theory, at least if our theory of the real numbers is categorical). This is a useful convention for discussion, at least. The possible use of non-separable Hilbert spaces in theoretical physics has provoked some inconclusive debate. In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist. ... In mathematics, descriptive set theory is the study of certain classes of well-behaved sets of real numbers, e. ... Jean-Alexandre-Eugène Dieudonné (July 1, 1906 - November 29, 1992) was a French mathematician, known for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of mathematics, particularly in... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, predicting physical phenomena through a physical theory. There are three types of theories in physics; mainstream theories, proposed theories and fringe theories. ...

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. Such constructive proofs can be turned into algorithms for use in numerical analysis, and they are the only sorts of proofs acceptable in constructive analysis. A famous example of a theorem of this sort is the Hahn-Banach theorem. Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... Flowcharts are often used to represent algorithms. ... In mathematics, the Hahn-Banach theorem is a central tool in functional analysis. ...

Examples

• A discrete space is separable iff it is countable.

• The real numbers are separable; they have the rational numbers as a countable dense subset. More generally, Euclidean space Rⁿ is separable, as the set of all points with rational coordinates is dense.

• The space of continuous functions on the unit interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (this is the Weierstrass approximation theorem).

• A Hilbert space is separable if and only if it has a countable orthonormal basis.

• An example of a separable space that is not second-countable is R_llt, the set of real numbers equipped with the lower limit topology.

• The product topology on the set of all functions (not necessarily continuous) from the real line to itself is a separable Hausdorff space. This space has cardinality 2^c, showing that separable spaces can still be rather "large". However, for separable Hausdorff spaces this is the largest possible cardinality. Note that this space is not first-countable.

• The trivial topology on any set is separable since any singleton is dense. This shows that by removing the Hausdorff requirement in the previous example we can get separable spaces with arbitrarily large cardinality.

In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... The text or formatting below is generated by a template which has been proposed for deletion. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematical analysis, a sequence { fn } of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, an orthonormal basis of an inner product space V(i. ... In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... The cardinality of a set is a property that describes the size of the set by describing it using a cardinal number. ... In set theory and other branches of mathematics, ‭ב‬2 (pronounced beth two), or 2c (pronounced two to the power of c), is a certain cardinal number. ... In topology, a first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ... In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ... Generally, a singleton is something which exists alone in some way. ...

Properties

• The continuous image of a separable space is separable. It follows that separability is a topological property preserved by homeomorphisms. It also follows that every quotient of a separable space is separable.

• A subspace of a separable space need not be separable, but every open subspace of a separable space is separable. Also every subspace of a separable metric space is separable.

• Every countable product of separable spaces is separable. Arbitrary products of separable spaces need not be separable.

• A separable, Hausdorff space X has cardinality less than or equal to 2^c (where c is the cardinality of the real numbers). If X is also first-countable then its cardinality is less than or equal to c.

• The set of all real-valued continuous functions on a separable space has a cardinality less than or equal to c. This follows since such functions are determined by their values on dense subsets.

• Every second-countable space is separable.

• A metric space is separable iff it is second-countable and iff it is Lindelöf.

• A separable uniform space whose uniformity has a countable basis is second-countable.