Separable space

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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.

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.

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.

Examples

  • An example of a separable space that is not second-countable is Rllt, 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 2c, 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.

Properties

  • A subspace of a separable space need not be separable (see the Sorgenfrey plane), 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.
  • 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.

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