set of all real-valued continuous functions on a separable space has a cardinality equal to , the cardinality of the continuum. This follows since such functions are determined by their values on dense subsets. From the above property, one can deduce the following: If X is a separable space having an uncountable closed discrete subspace, then X cannot be normal. This shows that the Sorgenfrey plane is not normal. For a compact Hausdorff space X, the following are equivalent: Embedding separable metric spaces Every separable metric space is homeomorphic to a subset of the Hilbert cube. This is established in the proof of the Urysohn metrization theorem. Every separable metric space is isometric to a subset of the (non-separable) Banach space l∞ of all bounded real sequences with the supremum norm; this is known as the Fréchet embedding. Every separable metric space is isometric to a subset of C([0,1]), the separable Banach space of continuous functions [0,1] → R, with the supremum norm. This is due to Stefan Banach. Every separable metric space is isometric to a subset of the Urysohn universal space. For nonseparable spaces: A metric space of density equal to an infinite cardinal is isometric to a subspace of , the space of real continuous functions on the product of copies of the unit interval. References General topology Properties of topological spaces