Marczewski's Separability Theorem

This article is about a theorem of topology. The article Theorem of Marczewski-Szpilrajn provides information on another theorem by Edward Marczewski .

The Separabilitätssatz of Marczewski ( English Marczewski's separability theorem ) - as a set of Marczewski called - is a tenet of mathematical sub-region of the general topology . It goes back to a work by the Polish mathematician Edward Marczewski from 1947 and deals with the problem of the separability of the product of certain topological spaces .

Formulation of the sentence

The sentence can be summarized as follows:

Let there be a non-empty family of Hausdorff rooms , all of which should consist of two or more elements , and let it be ${\displaystyle (X_{i})_{i\in I}\;(I\neq \emptyset )}$

${\displaystyle X=\prod _{i\in I}X_{i}}$

their topological product .

Then:

The product space is separable if and only if each of the spaces is separable and if, moreover, the index set has at most the thickness of the continuum . ${\displaystyle X}$ ${\displaystyle X_{i}}$ ${\displaystyle I}$

Related sentence

A theorem that is closely related to Marczewski's Separability Theorem is the following, which some authors call the Hewitt – Marczewski – Pondiczery Theorem ( English Hewitt – Marczewski – Pondiczery theorem ):

If an infinite cardinal number and is the product of topological spaces and these spaces all contain dense subsets whose thickness is at most , then the product space in turn includes a dense subset whose thickness is at most . ${\displaystyle \tau }$ ${\displaystyle X=\prod _{i\in I}X_{i}}$ ${\displaystyle |I|\leq 2^{\tau }}$ ${\displaystyle X_{i}}$ ${\displaystyle \tau }$ ${\displaystyle X}$ ${\displaystyle \tau }$

Note on naming

Kenneth Allen Ross and Arthur Harold Stone attribute the separability theorem to the American mathematician Ralph Boas , whose work from 1944 - which Boas published under the pseudonym ES Pondiczery - also contains this result.

literature

• WW Comfort : A short proof of Marczewski's separability theorem . In: American Mathematical Monthly . tape 76 , 1969, p. 1041-1042 , doi : 10.2307 / 2317135 ( MR0248742 ).
• AA Gryzlov : On dense subsets of Tychonoff products . In: Topology and its Applications . tape 170 , 2014, pp. 86-95 ( MR3200391 ).
• 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 .
• Edwin Hewitt : A remark on density characters . In: Bulletin of the American Mathematical Society . tape 52 , 1946, pp. 641-643 ( MR0017329 ).
• E. Marczewski: Séparabilité et multiplication cartésienne des espaces topologiques . In: Fundamenta Mathematicae . tape 34 , 1947, pp. 127-143 ( MR0021680 ).
• ES Pondiczery : Power problems in abstract spaces . In: Duke Mathematical Journal . tape 11 , 1944, pp. 835-837 , doi : 10.1215 / S0012-7094-44-01171-3 ( MR0011104 ).
• KA Ross , AH Stone : Products of separable spaces . In: American Mathematical Monthly . tape 71 , 1964, pp. 398-403 , doi : 10.2307 / 2313241 ( MR0164314 ).
• Stephen Willard : General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley , Reading, Massachusetts (et al.) 1970, pp. 224 ff . ( MR0264581 ).

See also

• Article "Separable space" in the English language Wikipedia
• Link to Hewitt-Marczewski-Pondiczery theorem (planetmath.org)

Individual evidence

1. ^ WW Comfort: A short proof of Marczewski's separability theorem. Amer. Math. Monthly 76, pp. 1041 ff
2. ↑ ^{a b c} J. Heine: Topology and Functional Analysis. 2002, p. 157
3. ↑ Stephen Willard: General Topology. 1978, p. 109
4. ↑ AA Gryzlov: On dense subsets of Tychonoff products. , Topology Appl. 170, p. 86 ff
5. ^ KA Ross, AH Stone: Products of separable spaces. , Amer. Math. Monthly 71, p. 399