Home
Hostname: page-component-564cf476b6-s5ssh Total loading time: 0.225 Render date: 2021-06-20T20:05:21.354Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

# Partitioning large vector spaces

Published online by Cambridge University Press:  12 March 2014

Corresponding

## Extract

The theme of this paper is the generalization of theorems about partitions of the sets of points and lines of finite-dimensional Euclidean spaces ℝ d to vector spaces over ℝ of arbitrary dimension and, more generally still, to arbitrary vector spaces over other fields so long as these fields are not too big. These theorems have their origins in the following striking theorem of Sierpiński [12] which appeared a half century ago.

Sierpiński's Theorem. The Continuum Hypothesis is equivalent to: There is a partition {X, Y, Z} of3 such that if ℓ is a line parallel to the x-axis [respectively: y-axis, z-axis] then X [respectively: Yℓ, Z] is finite.

The history of this theorem and some of its subsequent developments are discussed in the very interesting article by Simms [13]. Sierpiński's Theorem was generalized by Kuratowski [9] to partitions of ℝ n+2 into n + 2 sets obtaining an equivalence with . The geometric character that Sierpiński's Theorem and its generalization by Kuratowski appear to have is bogus, since the lines parallel to coordinate axes are essentially combinatorial, rather than geometric, objects. The following version of Kuratowski's theorem emphasizes its combinatorial character.

Kuratowski's Theorem. Let n < ω and A be any set. ThenA∣ ≤ ℵ n if and only if there is a partition P: A n+2n + 2 such that if in + 1 and ℓ is a line parallel to the i-th coordinate axis, then {x: P(x) = i} is finite.

Type
Research Article
Information
The Journal of Symbolic Logic , December 2003 , pp. 1171 - 1180

## Access options

Get access to the full version of this content by using one of the access options below.

## References

[1] Davies, R. O., On a denumerable partition problem of Erdős, Proceedings of the Cambridge Philosophical Society, vol. 59 (1963), pp. 501502.CrossRefGoogle Scholar
[2] Davies, R. O., On a problem of Erdős concerning decompositions of the plane, Proceedings of the Cambridge Philosophical Society, vol. 59 (1963), pp. 3336.CrossRefGoogle Scholar
[3] Davies, R. O., The power of the continuum and some propositions of plane geometry, Fundamenta Mathematicae, vol. 52 (1963), pp. 277281.CrossRefGoogle Scholar
[4] Erdős, P., Some remarks on set theory, III, Michigan Mathematics Journal, vol. 2 (1954), pp. 5157.Google Scholar
[5] Galvin, F., an apparently unpublished result from [7].Google Scholar
[6] Gruenhage, G., Covering properties on X2 ∖ Δ, W-sets, and compact subsets of Σ-products, Topology and its Applications, vol. 17 (1984), pp. 287304.CrossRefGoogle Scholar
[7] Komjáth, P., Set theoretic constructions in Euclidean spaces, Chapter XII in New trends in discrete and computational geometry (Pach, J., editor), Springer-Verlag, Berlin-Heidelberg-New York, 1993, pp. 303325.CrossRefGoogle Scholar
[8] Komjáth, P., Three clouds cover the plane, Annals of Pure and Applied Logic, vol. 109 (2001), pp. 7175.CrossRefGoogle Scholar
[9] Kuratowski, C., Sur une caracterisation des alephs, Fundamenta Mathematicae, vol. 38 (1951), pp. 1417.CrossRefGoogle Scholar
[10] Schmerl, J. H., Countable partitions of the sets of points and lines, Fundamenta Mathematicae, vol. 160 (1999), pp. 183196.Google Scholar
[11] Schmerl, J. H., How many clouds cover the plane?, Fundamenta Mathematicae, to appear.Google Scholar
[12] Sierpiński, W., Sur une propriété paradoxale de l'espace à trois dimensions équivalente à l'hypothèse du continu, Rendiconti del Circolo Matematico di Palermo (Serie II), vol. 1 (1952), pp. 710.CrossRefGoogle Scholar
[13] Simms, J. C., Sierpiński's theorem, Bulletin of the Belgian Mathematical Society, Simon Stevin, vol. 65 (1991), pp. 69163.Google Scholar

# Send article to Kindle

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Partitioning large vector spaces
Available formats
×

# Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Partitioning large vector spaces
Available formats
×

# Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Partitioning large vector spaces
Available formats
×
×