Separate room

The title of this article is ambiguous. Separable rooms are not identical to separated rooms .

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:

• The Banach space of bounded (real or complex) sequences is not separable. ${\displaystyle \ell ^{\infty }}$
• In general, for an infinite set the Banach space of bounded (real or complex valued) functions is never separable. ${\displaystyle S}$ ${\displaystyle B(X,{\mathcal {P}}(S))}$
• The space of almost periodic functions is a non-separable Hilbert space . ${\displaystyle \mathrm {AP} ^{2}}$
• If one provides the smallest uncountable ordinal number with its order topology, one obtains a non-separable space. ${\displaystyle \omega _{1}}$

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 ] .

literature

• PS Alexandroff : Introduction to set theory and general topology (=  university books for mathematics . Volume 85 ). VEB Deutscher Verlag der Wissenschaften , Berlin 1984.
• Thorsten Camps, Stefan Kühling, Gerhard Rosenberger: Introduction to set-theoretical and algebraic topology (=  Berlin study series on mathematics . Volume 15 ). Heldermann Verlag, Lemgo 2006, ISBN 3-88538-115-X .
• Charles O. Christenson, William L. Voxman: Aspects of Topology . 2nd Edition. BCS Associates, Moscow, Idaho, USA 1998, ISBN 0-914351-08-7 .
• Lutz Führer : General topology with applications . Vieweg Verlag , Braunschweig 1977, ISBN 3-528-03059-3 .
• Leszek Gasiński, Nikolaos S. Papageorgiou: Exercises in Analysis. Part 1 (=  Problem Books in Mathematics ). Springer-Verlag , Cham, Heidelberg, New York, Dordrecht, London 2014, ISBN 978-3-319-06175-7 , doi : 10.1007 / 978-3-319-06176-4 .
• Jürgen Heine: Topology and Functional Analysis . Basics of abstract analysis with applications. 2nd, improved edition. Oldenbourg Verlag , Munich 2011, ISBN 978-3-486-70530-0 .
• GJO Jameson: Topology and Normed Spaces . Chapman and Hall, London 1974, ISBN 0-412-12880-2 .
• Joseph Muscat : Functional Analysis . An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras. Springer-Verlag, Cham, Heidelberg, New York, Dordrecht, London 2014, ISBN 978-3-319-06727-8 , doi : 10.1007 / 978-3-319-06728-5 .
• Mark Neumark : Normalized Algebras . Verlag Harri Deutsch , Thun and Frankfurt / Main 1990, ISBN 3-8171-1001-4 .
• Horst Schubert: Topology (=  mathematical guidelines ). 4th edition. BG Teubner , Stuttgart 1975, ISBN 3-519-12200-6 .
• Lynn A. Steen , J. Arthur Seebach, Jr.,: Counterexamples in Topology . 2nd Edition. Springer-Verlag, New York, Heidelberg, Berlin 1978, ISBN 0-387-90312-7 .
• Dirk Werner : Functional Analysis (=  Springer textbook ). 6th, corrected edition. Springer-Verlag, Berlin, Heidelberg, New York 2007, ISBN 978-3-540-72533-6 .
• Stephen Willard: General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley , Reading, Massachusetts et al. a. 1970.

See also

• Countable amount
• Hausdorffraum
• Hilbert dream
• Metric space
• Compact space
• Polish area
• Second axiom of countability

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.