Abstract
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 http://dx.doi.org/10.1515/JAA.1995.18110.1515/JAA.1995.181Search 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 http://dx.doi.org/10.1016/0001-8708(84)90026-410.1016/0001-8708(84)90026-4Search 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 http://dx.doi.org/10.1007/s00153005009610.1007/s001530050096Search 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 http://dx.doi.org/10.2307/227584110.2307/2275841Search 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
© 2013 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
