Editing Separable space (section)

==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.