# Separate room

The mathematical term separable describes a frequently used countability property of topological spaces in topology and related areas . The term is of particular importance in functional analysis . Here, for example, one can show that there are always countable orthonormal bases in a separable Hilbert space .

## definition

A topological space is called separable if there is at most a countable subset that is dense in this space .

## Criteria for separable rooms

• If a topological space has a (at most) countable basis , it is separable. (The reverse is generally not true.)
• For a metric space the following even applies: ${\ displaystyle X}$
• For it to have a countable basis, it is necessary and sufficient that it is separable.${\ displaystyle X}$${\ displaystyle X}$
• A totally restricted metric space is always separable.${\ displaystyle X}$
• In particular, each compact , metrizable room can be separated. The following applies more precisely:
• If a metrizable topological space is, then the three properties are ${\ displaystyle X}$
• (1) have a countable base,
• (2) to be lindelöfsch ,
• (3) to be separable,
equivalent to.
• A topological vector space is separable if and only if there is a countable subset, so that the subspace generated by it is dense.
• If a Hilbert space is of infinite dimension , then the following three conditions are always equivalent: ${\ displaystyle X}$
• (1) is separable.${\ displaystyle X}$
• (2) All orthonormal bases of are countable.${\ displaystyle X}$
• (3) In there is a countable orthonormal basis.${\ displaystyle X}$
• For an infinite and linearly ordered set with the order topology , the following three conditions are always equivalent: ${\ displaystyle X}$
• (1) is separable and contiguous .${\ displaystyle X}$
• (2) is order isomorphic to an interval of .${\ displaystyle X}$${\ displaystyle \ mathbb {R}}$
• (3) is homeomorphic to an interval of .${\ displaystyle X}$${\ displaystyle \ mathbb {R}}$
• If a metric space is connected and locally Euclidean , then it is lindelöfsch and thus separable.${\ displaystyle X}$

## Examples

Examples of separable rooms are:

• The rooms are separable because they can be counted and are located close together .${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle \ mathbb {Q} ^ {n}}$${\ displaystyle \ mathbb {R} ^ {n}}$
• The rooms with a restricted, open subset and are separable.${\ displaystyle L ^ {p} (\ Omega)}$${\ displaystyle \ Omega \ subset \ mathbb {R} ^ {n}}$${\ displaystyle 1 \ leq p <\ infty}$
• The sequence spaces for are separable.${\ displaystyle \ ell ^ {p}}$${\ displaystyle 1 \ leq p <\ infty}$
• The space of the ( real or complex ) null sequences is a separable Banach space with the supremum norm .${\ displaystyle c_ {0}}$
• The space of the terminating sequences ( ) is separable with the norm for .${\ displaystyle c_ {00}}$${\ displaystyle \ forall x \ in c_ {00} \ exists N \ in \ mathbb {N} \ forall n \ geq N: x_ {n} = 0}$${\ displaystyle \ ell ^ {p}}$${\ displaystyle 1 \ leq p <\ infty}$
• The spaces are always separable for open subsets and natural numbers .${\ displaystyle \ Omega \ subseteq \ mathbb {R} ^ {n}}$ ${\ displaystyle k}$${\ displaystyle C ^ {k} (\ Omega)}$
• Every infinite set with a cofinite topology is separable because any countably infinite subset has the entire space as the only closed superset.
• The Niemytzki plane (or Moore plane) is a separable space, since the countable (!) Subset of points with rational coordinates it contains is dense.

## Counterexamples

There are some well-known examples of non-separable spaces:

## Permanent properties

• Images of separable rooms under continuous functions are separable again.
• Open sub-spaces of separable spaces are always also separable.
• In general, subspaces of separable spaces are not separable. For example, the aforementioned separable (!) Niemytzki level contains a nonseparable subspace.
• It is true, however, that subspaces of separable metric spaces are separable again.
• Marczewski's theorem of separability : If a family of separable spaces and if the thickness is at most equal to the thickness of the continuum , then it is also separable with the product topology . To see this result, it suffices to prove the separability of . To do this, it is easy to consider that the countable set of finite sums of functions from is dense, where is the characteristic function of the interval .${\ displaystyle (X_ {i}) _ {i \ in I}}$${\ displaystyle I}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ textstyle \ prod _ {i \ in I} X_ {i}}$${\ displaystyle {\ mathbb {N}} ^ {\ mathbb {R}} = \ {f \ mid f \ colon {\ mathbb {R}} \ rightarrow {\ mathbb {N}} \}}$${\ displaystyle \ {n \ cdot \ chi _ {[a, b]}; n \ in {\ mathbb {N}}, a, b \ in {\ mathbb {Q}} \}}$${\ displaystyle \ chi _ {[a, b]}}$${\ displaystyle [a, b]}$

## Connection with other terms

• In the English specialist literature, a topological space with (at most) a countable base is described by some authors as completely separable or perfectly separable , i.e. as completely separable or completely separable .${\ displaystyle X}$
• If the topology of a separable space can be generated by a complete metric , it is called a Polish space .${\ displaystyle X}$ ${\ displaystyle X}$
• The concept of separable space has no relation to the concept of separated space .

## To the history

• The concept of the separable space goes back to Maurice René Fréchet and his publication Sur quelques points de calcul fonctionnel from 1906.
• According to PS Alexandroff, the term separable is a most unfortunate term ... which, unfortunately, has become naturalized and widespread.
• As Horst Schubert wrote in 1975, there were ... tendencies to abolish it [the term separable ] .

## Individual evidence

1. Thorsten Camps, Stefan Kühling, Gerhard Rosenberger: Introduction to set-theoretical and algebraic topology. 2006, p. 34.
2. ^ PS Alexandroff: Introduction to set theory and general topology. 1984, p. 121.
3. ^ Joseph Muscat: Functional Analysis. 2014, p. 68.
4. Leszek Gasiński, Nikolaos S. Papageorgiou: Exercises in Analysis. Part 1. 2014, p. 8.
5. Since compactness is a special case of the Lindelöf property, the aforementioned statement results from this equivalence as a consequence.
6. Jürgen Heine: Topology and Functional Analysis. 1970, p. 261.
7. Dirk Werner: Functional Analysis. 2007, p. 235.
8. ^ Lutz Führer: General topology with applications. 1977, p. 129.
9. ^ Charles O. Christenson, William L. Voxman: Aspects of Topology. 1998, p. 420.
10. a b Heine, op.cit, p. 72.
11. ^ GJO Jameson: Topology and Normed Spaces. 1970, p. 159.
12. Camps / Kühling / Rosenberger, op.cit, p. 18.
13. a b Lynn A. Steen, J. Arthur Seebach, Jr.,: Counterexamples in Topology. 1970, p. 7, pp. 100-103.
14. Heine, op.cit, p. 86.
15. Heine, op.cit, p. 72.
16. Jameson, op.cit, p. 158.
17. Stephen Willard: General Topology. 1970, p. 114.
18. The image of the dense subset in the domain of definition simply serves as the dense subset in the image.
19. Führer, op.cit, p. 128.
20. This follows from the equivalence mentioned above, because the latter is obviously carried over to the metric subspaces.
21. Steen / Seebach, op.cit, p. 162.
22. Gasiński / Papageorgiou, op.cit, p. 226.
23. a b Horst Schubert: Topology. 1975, p. 58.
24. Willard, op.cit, p. 303.
25. Alexandroff, op. Cit, pp. 120-121.
26. Which apparently did not happen.