the only sorts of proofs acceptable in constructive analysis. A famous example of a theorem of this sort is the Hahn–Banach theorem. Further examples Separable spaces Every compact metric space (or metrizable space) is separable. Any topological space that is the union of a countable number of separable subspaces is separable. Together, these first two examples give a different proof that -dimensional Euclidean space is separable. The space of all continuous functions from a compact subset to the real line is separable. The Lebesgue spaces , over a separable measure space , are separable for any . The space of continuous real-valued functions on the unit interval with the metric of uniform convergence is a separable space, since it follows from the Weierstrass approximation theorem that the set of polynomials in one variable with rational coefficients is a countable dense subset of . The Banach–Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linear subspace of . A Hilbert space is separable if and only if it has a countable orthonormal basis. It follows that any separable, infinite-dimensional Hilbert space is isometric to the space of square-summable sequences. An example of a separable space that is not second-countable is the Sorgenfrey line , the set of real numbers equipped with the lower limit topology. A separable σ-algebra is a σ-algebra that is a separable space when considered as a metric space with metric for and a given measure (and with being the symmetric difference operator). Non-separable spaces The first uncountable ordinal , equipped with its natural order topology, is not separable. The Banach space of all bounded real sequences, with the supremum norm, is not separable. The same holds for . The Banach space of functions of bounded variation is not separable; note however that this space has very important applications in mathematics, physics and engineering. Properties A subspace of a separable space need not be separable (see the Sorgenfrey plane and the Moore plane), but every open subspace of a separable space is separable . Also every subspace of a separable metric space is separable. In fact, every topological space is a subspace of a separable space of the same cardinality. A construction adding at most countably many points is given in ; if the space was a Hausdorff space then the space constructed that it embeds into is also a Hausdorff space. The