Skip to main content Accessibility help
×
Home
Hostname: page-component-684bc48f8b-vgwqb Total loading time: 15.23 Render date: 2021-04-14T04:31:04.880Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Analytic countably splitting families

Published online by Cambridge University Press:  12 March 2014

Otmar Spinas
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität Zu Kiel, Ludewig-Meyn-Straße 4, 24098 Kiel, Germany, E-mail: spinas@math.uni-kiel.de
Corresponding
E-mail address:

Abstract

A family A(ω) is called countably splitting if for every countable F ⊆ [ω]ω, some element of A splits every member of F. We define a notion of a splitting tree, by means of which we prove that every analytic countably splitting family contains a closed countably splitting family. An application of this notion solves a problem of Blass. On the other hand we show that there exists an Fσ splitting family that does not contain a closed splitting family.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2004

Access options

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

References

[1]Blass, A., Reductions between cardinal characteristics of the continuum, Set theory (Annual Boise extravaganza in set theory conference, 1992–94) (Bartoszyński, T. and Scheepers, M., editors), Contemporary Mathematics, vol. 192, American Mathematical Society, 1996, pp. 3149.Google Scholar
[2]Blass, A., Combinatorial cardinal characteristics of the continuum. Handbook of Set Theory, to appear.Google Scholar
[3]Brendle, J., Hjorth, G., and Spinas, O., Regularity properties for dominating projective sets. Annals of Pure and Applied Logic, vol. 72 (1995), pp. 291307.CrossRefGoogle Scholar
[4]Davis, M., Infinite games of perfect information. Annals of Mathematics Studies, vol. 52 (1964), pp. 85101.Google Scholar
[5]Kechris, A. S., On a notion of smallness for subsets of the Baire space. Transactions of the American Mathematical Society, vol. 229 (1977), pp. 191207.CrossRefGoogle Scholar
[6]Kechris, A. S., Classical descriptive set theory, Springer-Verlag, New York. 1995.CrossRefGoogle Scholar
[7]Kunen, K., Set theory, An introduction to independence proofs, North-Holland Publishing Co.. Amsterdam, 1983.Google Scholar
[8]Kuratowski, K., Topology, vol. 1, Academic Press, 1980.Google Scholar
[9]Martin, D., Borel determinacy. Annals of Mathematics, II, vol. 102 (1975), pp. 363371.CrossRefGoogle Scholar
[10]Mildenberger, H., Non-constructive Galois-Tukey connections, this Journal, vol. 62 (1997), pp. 11791187.Google Scholar
[11]Martin, D., No Borel connections for the unsplitting relations. Mathematical Logic Quarterly, vol. 48 (2002), pp. 517521.Google Scholar
[12]Raymond, J. Saint, Approximation des sous-ensembles analytiques par l'intérieur, Comptes Rendus de l'Académie des Sciences Paris, Série A, vol. 281 (1975), pp. 8587.Google Scholar
[13]Spinas, O., Dominating projective sets in the Baire space. Annals of Pure and Applied Logic, vol. 68 (1994). pp. 327342.CrossRefGoogle Scholar
[14]Vojtáš, P., Generalized Galois-Tukey connections between relations on classical objects of real analysis, Set theory of the reals (Judah, H., editor), Israel Mathematical Conference Proceedings, vol. 6, American Mathematical Society, 1993, pp. 619643.Google Scholar
[15]Yiparaki, O., On some tree partitions, Ph.D. thesis, University of Michigan, 1994.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 8 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 14th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your 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.

Analytic countably splitting families
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.

Analytic countably splitting families
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.

Analytic countably splitting families
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *