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'.