Editing Separable space (section)

===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 [[Isometry|isometric]] to a subset of the (non-separable) [[Banach space]] ''l''<sup>∞</sup> of all bounded real sequences with the [[uniform norm|supremum norm]]; this is known as the Fréchet embedding. {{harv | Heinonen | 2003}}
* Every separable metric space is isometric to a subset of C([0,1]), the separable Banach space of continuous functions [0,1]&nbsp;→&nbsp;'''R''', with the [[uniform norm|supremum norm]]. This is due to [[Stefan Banach]]. {{harv | Heinonen | 2003}}
* Every separable metric space is isometric to a subset of the [[Urysohn universal space]].
''For nonseparable spaces'':
* A [[metric space]] of [[dense set|density]] equal to an infinite cardinal {{mvar|α}} is isometric to a subspace of {{math|C([0,1]<sup>α</sup>, '''R''')}}, the space of real continuous functions on the product of {{mvar|α}} copies of the unit interval. {{harv|Kleiber|Pervin|1969}}