Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 22, 2012

On the structure of perfect sets in various topologies associated with tree forcings

  • Andrzej Nowik EMAIL logo and Patrick Reardon
From the journal Open Mathematics


We prove that the Ellentuck, Hechler and dual Ellentuck topologies are perfect isomorphic to one another. This shows that the structure of perfect sets in all these spaces is the same. We prove this by finding homeomorphic embeddings of one space into a perfect subset of another. We prove also that the space corresponding to eventually different forcing cannot contain a perfect subset homeomorphic to any of the spaces above.

[1] Balcerzak M., Rosłanowski A., Coinitial families of perfect sets, J. Appl. Anal., 1995, 1(2), 181–204 in Google Scholar

[2] Carlson T.J., Simpson S.G., A dual form of Ramsey’s theorem, Adv. in Math., 1984, 53(3), 265–290 in Google Scholar

[3] van Douwen E.K., The Pixley-Roy topology on spaces of subsets, In: Set-Theoretic Topology, Athens, Ohio, 1975–1976, Academic Press, New York-London, 1977, 111–134 10.1016/B978-0-12-584950-0.50015-8Search in Google Scholar

[4] Halbeisen L., Symmetries between two Ramsey properties, Arch. Math. Logic, 1998, 37(4), 241–260 in Google Scholar

[5] Łabędzki G., A topology generated by eventually different functions, Acta Univ. Carolin. Math. Phys., 1996, 37(2), 37–53 Search in Google Scholar

[6] Łabędzki G., Repický M., Hechler reals, J. Symbolic Logic, 1995, 60(2), 444–458 in Google Scholar

[7] Nowik A., Reardon P., A dichotomy theorem for the Ellentuck topology, Real Anal. Exchange, 2003/04, 29(2), 531–542 10.14321/realanalexch.29.2.0531Search in Google Scholar

[8] Płotka K., Recław I., Finitely continuous Hamel functions, Real Anal. Exchange, 2004/05, 30(2), 867–870 10.14321/realanalexch.30.2.0867Search in Google Scholar

[9] Popov V., On the subspaces of exp X, In: Topology, Vol. 2, Budapest, August 7–11, 1978, Colloq. Math. Soc. János Bolyai, 23, North-Holland, Amsterdam-New York, 1980, 977–984 Search in Google Scholar

[10] Reardon P., Ramsey, Lebesgue, and Marczewski sets and the Baire property, Fund. Math., 1996, 149(3), 191–203 10.4064/fm-149-3-191-203Search in Google Scholar

Published Online: 2012-12-22
Published in Print: 2013-3-1

© 2013 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 23.3.2023 from
Scroll Up Arrow