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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures


Authors: Tomek Bartoszynski and Boaz Tsaban
Journal: Proc. Amer. Math. Soc. 134 (2006), 605-615
MSC (2000): Primary 54G20, 54G15, 54D20
Published electronically: June 29, 2005
MathSciNet review: 2176030
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Abstract: We consider the question of which of the major classes defined by topological diagonalizations of open or Borel covers is hereditary. Many of the classes in the open case are not hereditary already in ZFC, and none of them are provably hereditary. This is in contrast with the Borel case, where some of the classes are provably hereditary. Two of the examples are counter-examples of sizes $\mathfrak{d}$ and $\mathfrak{b}$, respectively, to the Menger and Hurewicz Conjectures, and one of them answers a question of Steprans on perfectly meager sets.


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Additional Information

Tomek Bartoszynski
Affiliation: Department of Mathematics and Computer Science, Boise State University, Boise, Idaho 83725
Email: tomek@math.boisestate.edu

Boaz Tsaban
Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel
Address at time of publication: Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel
Email: tsaban@macs.biu.ac.il, boaz.tsaban@weizmann.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07997-9
PII: S 0002-9939(05)07997-9
Keywords: Selection principles, strong $\gamma$-set, Menger property, Hurewicz property
Received by editor(s): January 4, 2004
Received by editor(s) in revised form: September 20, 2004
Published electronically: June 29, 2005
Additional Notes: The first author was partially supported by NSF grant DMS 0200671.
This paper constitutes a part of the second author’s doctoral dissertation at Bar-Ilan University.
Communicated by: Alan Dow
Article copyright: © Copyright 2005 American Mathematical Society