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Separable

In mathematics, a metric space is separable if it has a countable dense subset. Most of the examples initially encountered are indeed separable: for example the real numbers with their standard metric have the rational numbers as a countable dense subset. Since the space of continuous functions[?] on the interval [0,1] has a dense subset of polynomials, for the uniform metric, and their coefficients can be approximated by rationals, that space is also separable. A Hilbert space is separable if and only if it has a countable orthogonal basis.

For technical reasons the foundations of general topology are written to remove the requirement of separability, or other 'axioms of countability'.



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