Editing Separable space (section)

==Properties==
* A [[subspace (topology)|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 {{harv|Willard|1970|loc=Th 16.4b}}. 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 {{harv|Sierpiński|1952|p=49}}; if the space was a Hausdorff space then the space constructed that it embeds into is also a Hausdorff space.
* The set of all real-valued continuous functions on a separable space has a cardinality equal to <math>\mathfrak{c}</math>, 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 space|normal]]. This shows that the [[Sorgenfrey plane]] is not normal.
*For a [[compact space|compact]] [[Hausdorff space]] ''X'', the following are equivalent: {{ ordered list | list-style-type = lower-roman
| 1 = ''X'' is second countable.
| 2 = The space <math>\mathcal{C}(X,\mathbb{R})</math> of continuous real-valued functions on ''X'' with the [[uniform norm|supremum norm]] is separable.
| 3 = ''X'' is metrizable.}}

===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}}