Editing Separable space

{{Short description|Topological space with a dense countable subset}}{{distinguish|Separated space|Separation axiom}}

In [[mathematics]], a [[topological space]] is called '''separable''' if it contains a [[countable set|countable]], [[dense (topology)|dense]] subset; that is, there exists a [[sequence]] <math>( x_n )_{n=1}^{\infty} </math> of elements of the space such that every nonempty [[open subset]] of the space contains at least one element of the sequence.

Like the other [[axioms of countability]], separability is a "limitation on size", not necessarily in terms of [[cardinality]] (though, in the presence of the [[Hausdorff space|Hausdorff axiom]], this does turn out to be the case; see below) but in a more subtle topological sense.  In particular, every [[continuous function]] on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.

Contrast separability with the related notion of [[second countability]], which is in general stronger but equivalent on the class of [[metrizable]] spaces.

==First examples==
Any topological space that is itself [[finite set|finite]] or [[countably infinite]] is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the [[real line]], in which the [[rational numbers]] form a countable dense subset. Similarly the set of all length-<math>n</math> [[Vector (mathematics and physics)|vectors]] of rational numbers, <math>\boldsymbol{r}=(r_1,\ldots,r_n) \in \mathbb{Q}^n</math>, is a countable dense subset of the set of all length-<math>n</math> vectors of real numbers, <math>\mathbb{R}^n</math>; so for every <math>n</math>, <math>n</math>-dimensional [[Euclidean space]] is separable.

A simple example of a space that is not separable is a [[discrete space]] of uncountable cardinality.

Further examples are given below.

==Separability versus second countability==
Any [[second-countable space]] is separable: if <math>\{U_n\}</math> is a countable base, choosing any <math>x_n \in U_n</math> from the non-empty <math>U_n</math> gives a countable dense subset. Conversely, a [[metrizable space]] is separable if and only if it is second countable, which is the case if and only if it is [[Lindelöf space|Lindelöf]].

To further compare these two properties:
* An arbitrary [[subspace (topology)|subspace]] of a second-countable space is second countable; subspaces of separable spaces need not be separable (see below).
* Any continuous image of a separable space is separable {{harv|Willard|1970|loc=Th. 16.4a}}; even a [[quotient topology|quotient]] of a second-countable space need not be second countable.
* A [[product topology|product]] of at most continuum many separable spaces is separable {{harv | Willard | 1970 | loc=Th 16.4c | p=109 }}. A countable product of second-countable spaces is second countable, but an uncountable product of second-countable spaces need not even be first countable.

We can construct an example of a separable topological space that is not second countable.  Consider any uncountable set <math>X</math>, pick some <math>x_0 \in X</math>, and define the topology to be the collection of all sets that contain <math>x_0</math> (or are empty).  Then, the closure of <math>{x_0}</math> is the whole space (<math>X</math> is the smallest closed set containing <math>x_0</math>), but every set of the form <math>\{x_0, x\}</math> is open.  Therefore, the space is separable but there cannot have a countable base.

==Cardinality==
The property of separability does not in and of itself give any limitations on the [[cardinality]] of a topological space: any set endowed with the [[trivial topology]] is separable, as well as second countable, [[quasi-compact]], and [[connected space|connected]]. The "trouble" with the trivial topology is its poor separation properties: its [[Kolmogorov quotient]] is the one-point space.

A [[first-countable]], separable Hausdorff space (in particular, a separable metric space) has at most the [[cardinality of the continuum|continuum cardinality]] <math>\mathfrak{c}</math>. In such a space, [[Closure (topology)|closure]] is determined by limits of sequences and any convergent sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of <math>X</math>.

A separable Hausdorff space has cardinality at most <math>2^\mathfrak{c}</math>, where <math>\mathfrak{c}</math> is the cardinality of the continuum. For this closure is characterized in terms of [[Filters in topology|limits of filter bases]]: if <math>Y\subseteq X</math> and <math>z\in X</math>, then <math>z\in\overline{Y}</math> if and only if there exists a filter base <math>\mathcal{B}</math> consisting of subsets of <math>Y</math> that converges to <math>z</math>. The cardinality of the set <math>S(Y)</math> of such filter bases is at most <math>2^{2^{|Y|}}</math>. Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection <math>S(Y) \rightarrow X</math> when <math>\overline{Y}=X.</math>

The same arguments establish a more general result: suppose that a Hausdorff topological space <math>X</math> contains a dense subset of cardinality <math>\kappa</math>.
Then <math>X</math> has cardinality at most <math>2^{2^{\kappa}}</math> and cardinality at most <math>2^{\kappa}</math> if it is first countable.

The product of at most continuum many separable spaces is a separable space {{harv | Willard | 1970 | loc=Th 16.4c | p=109 }}. In particular the space <math>\mathbb{R}^{\mathbb{R}}</math> of all functions from the real line to itself, endowed with the product topology, is a separable Hausdorff space of cardinality <math>2^\mathfrak{c}</math>. More generally, if <math>\kappa</math> is any infinite cardinal, then a product of at most <math>2^\kappa</math> spaces with dense subsets of size at most <math>\kappa</math> has itself a dense subset of size at most <math>\kappa</math> ([[Hewitt–Marczewski–Pondiczery theorem]]).

==Constructive mathematics==
Separability is especially important in [[numerical analysis]] and [[Mathematical constructivism|constructive mathematics]], since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs can be turned into [[algorithm]]s for use in numerical analysis, and they are 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 <math>n</math>-dimensional Euclidean space is separable.
* The space <math>C(K)</math> of all continuous functions from a [[Compact space|compact]] subset <math>K\subseteq\mathbb{R}</math> to the real line <math>\mathbb{R}</math> is separable.
* The [[Lp space|Lebesgue spaces]] <math>L^{p}\left(X,\mu\right)</math>, over a measure space <math>\left\langle X,\mathcal{M},\mu\right\rangle</math>  whose σ-algebra is countably generated and whose measure is σ-finite, are separable for any <math>1\leq p<\infty</math>.<ref>{{cite book|author=[[Donald L. Cohn]]|date=2013|language=en|publisher=[[Springer Science+Business Media]]|title=Measure Theory|series=Birkhäuser Advanced Texts Basler Lehrbücher |doi=10.1007/978-1-4614-6956-8 |isbn=978-1-4614-6955-1 |url=https://link.springer.com/book/10.1007/978-1-4614-6956-8}}<!-- auto-translated by Module:CS1 translator -->, {{Lang|en|Proposition}} 3.4.5.</ref>
* The space <math>C([0,1])</math> of [[Continuous function|continuous real-valued functions]] on the [[unit interval]] <math>[0,1]</math> with the metric of [[uniform convergence]] is a separable space, since it follows from the [[Stone–Weierstrass theorem|Weierstrass approximation theorem]] that the set <math>\mathbb{Q}[x]</math> of polynomials in one variable with rational coefficients is a countable dense subset of <math>C([0,1])</math>. The [[Banach–Mazur theorem]] asserts that any separable [[Banach space]] is isometrically isomorphic to a closed [[linear subspace]] of <math>C([0,1])</math>.
* 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 <math>\ell^2</math> of square-summable sequences.
* An example of a separable space that is not second-countable is the [[Sorgenfrey line]] <math>\mathbb{S}</math>, the set of real numbers equipped with the [[lower limit topology]].
* A [[&sigma;-algebra#Separable &sigma;-algebras|separable &sigma;-algebra]] is a &sigma;-algebra <math>\mathcal{F}</math> that is a separable space when considered as a [[metric space]] with [[metric (mathematics)|metric]] <math>\rho(A,B) = \mu(A \triangle B)</math> for <math>A,B \in \mathcal{F}</math> and a given finite [[measure (mathematics)|measure]] <math>\mu</math> (and with <math>\triangle</math> being the [[symmetric difference]] operator).<ref>{{cite journal|last1=Džamonja|first1=Mirna|last2=Kunen|first2=Kenneth|author-link2=Kenneth Kunen|title=Properties of the class of measure separable compact spaces| journal=[[Fundamenta Mathematicae]]|year=1995|pages=262|url=https://archive.uea.ac.uk/~h020/fundamenta.pdf|quote=If <math>\mu</math> is a Borel measure on <math>X</math>, the measure algebra of <math>(X,\mu)</math> is the Boolean algebra of all Borel sets modulo <math>\mu</math>-null sets. If <math>\mu</math> is finite, then such a measure algebra is also a metric space, with the distance between the two sets being the measure of their symmetric difference. Then, we say that <math>\mu</math> is ''separable'' [[if and only if|iff]] this metric space is separable as a topological space.|bibcode=1994math......8201D|arxiv=math/9408201}}</ref>

===Non-separable spaces===
* The [[first uncountable ordinal]] <math>\omega_1</math>, equipped with its natural [[order topology]], is not separable.
* The [[Banach space]] <math>\ell^\infty</math> of all bounded real sequences, with the [[uniform norm|supremum norm]], is not separable. The same holds for  <math>L^\infty</math>.
* The [[Banach space]] of [[Bounded variation|functions of bounded variation]] is not separable; note however that this space has very important applications in mathematics, physics and engineering.

==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}}

==References==
{{Reflist}}
*{{Citation|title=Geometric embeddings of metric spaces|url=http://www.math.jyu.fi/research/reports/rep90.pdf|first=Juha|last=Heinonen|date=January 2003|access-date=6 February 2009}}
{{Use dmy dates|date=August 2019}}
*{{Citation | last1=Kelley | first1=John L. | author1-link=John L. Kelley | title=General Topology | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90125-1 | mr=0370454  | year=1975}}
*{{citation|last1=Kleiber|first1=Martin|last2=Pervin|first2=William J.|title=A generalized Banach-Mazur theorem|journal=Bull. Austral. Math. Soc.|date=1969|volume=1|issue=2|pages=169–173|doi=10.1017/S0004972700041411|doi-access=free}}
*{{Citation | last1=Sierpiński | first1=Wacław | author1-link=Wacław Sierpiński | title=General topology | publisher=University of Toronto Press | location=Toronto, Ont. | series=Mathematical Expositions, No. 7 | mr=0050870 | year=1952}}
*{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | orig-year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995| title-link=Counterexamples in Topology }}
*{{Citation | last1=Willard | first1=Stephen | title=General Topology | publisher=[[Addison-Wesley]] | isbn=978-0-201-08707-9 | mr=0264581 | year=1970 | url-access=registration | url=https://archive.org/details/generaltopology00will_0 }}

{{DEFAULTSORT:Separable Space}}
[[Category:General topology]]
[[Category:Properties of topological spaces]]