Point-cofinite covers in the Laver model

Authors:
Arnold W. Miller and Boaz Tsaban

Journal:
Proc. Amer. Math. Soc. **138** (2010), 3313-3321

MSC (2010):
Primary 03E35, 26A03; Secondary 03E17

DOI:
https://doi.org/10.1090/S0002-9939-10-10407-9

Published electronically:
April 30, 2010

MathSciNet review:
2653961

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the statement: For each sequence of point-cofinite open covers, one can pick one element from each cover and obtain a point-cofinite cover. is the minimal cardinality of a set of reals not satisfying . We prove the following assertions:

- If there is an unbounded tower, then there are sets of reals of cardinality satisfying .
- It is consistent that all sets of reals satisfying have cardinality smaller than .

The main technical result is that in Laver's model, each set of reals of cardinality has an unbounded Borel image in the Baire space .

**1.**T. Bartoszyński and S. Shelah,*Continuous images of sets of reals*, Topology and its Applications**116**(2001), 243-253. MR**1855966 (2002f:03090)****2.**T. Bartoszyński and B. Tsaban,*Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures*, Proceedings of the American Mathematical Society,**134**(2006), 605-615. MR**2176030 (2006f:54038)****3.**J. Baumgartner and R. Laver,*Iterated perfect-set forcing*, Annals of Mathematical Logic**17**(1979), 271-288. MR**556894 (81a:03050)****4.**A. Blass,*Combinatorial cardinal characteristics of the continuum*, in:**Handbook of Set Theory**(M. Foreman, A. Kanamori, and M. Magidor, eds.), Kluwer Academic Publishers, Dordrecht, to appear.`http://www.math.lsa.umich.edu/˜ablass/hbk.pdf`**5.**A. Dow,*Two classes of Fréchet-Urysohn spaces*, Proceedings of the American Mathematical Society**108**(1990), 241-247. MR**975638 (90j:54028)****6.**F. Galvin and A. Miller,*-sets and other singular sets of real numbers*, Topology and its Applications**17**(1984), 145-155. MR**738943 (85f:54011)****7.**J. Gerlits and Zs. Nagy,*Some properties of . I*, Topology and its Applications**14**(1982), 151-161. MR**667661 (84f:54021)****8.**W. Hurewicz,*Über eine Verallgemeinerung des Borelschen Theorems*, Mathematische Zeitschrift**24**(1925), 401-421.**9.**W. Just, A. Miller, M. Scheepers, and P. Szeptycki,*The combinatorics of open covers. II*, Topology and its Applications**73**(1996), 241-266. MR**1419798 (98g:03115a)****10.**L. Kočinac,*Selection principles related to -properties*, Taiwanese Journal of Mathematics**12**(2008), 561-572. MR**2417134 (2009h:91049)****11.**R. Laver,*On the consistency of Borel's conjecture*, Acta Mathematica**137**(1976), 151-169. MR**0422027 (54:10019)****12.**A. Miller,*Mapping a set of reals onto the reals*, Journal of Symbolic Logic**48**(1983), 575-584. MR**716618 (84k:03125)****13.**P. Nyikos,*Subsets of and the Fréchet-Urysohn and -properties*, Topology and its Applications**48**(1992), 91-116. MR**1195504 (93k:54011)****14.**M. Sakai,*The sequence selection properties of*, Topology and its Applications**154**(2007), 552-560. MR**2280899 (2007k:54007)****15.**M. Scheepers,*Combinatorics of open covers. I: Ramsey theory*, Topology and its Applications**69**(1996), 31-62. MR**1378387 (97h:90123)****16.**M. Scheepers,*and Arhangel'skiĭ's spaces*, Topology and its Applications**89**(1998), 265-275. MR**1645184 (99g:54018)****17.**M. Scheepers,*Sequential convergence in and a covering property*, East-West Journal of Mathematics**1**(1999), 207-214. MR**1727383 (2000i:54014)****18.**M. Scheepers and B. Tsaban,*The combinatorics of Borel covers*, Topology and its Applications**121**(2002), 357-382. MR**1908999 (2003e:03091)****19.**B. Tsaban,*Menger's and Hurewicz's Problems: Solutions from ``The Book'' and refinements*, Contemporary Mathematics, American Mathematical Society, to appear.**20.**B. Tsaban and L. Zdomskyy,*Hereditarily Hurewicz spaces and Arhangel'skiĭ sheaf amalgamations*, Journal of the European Mathematical Society, to appear.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
03E35,
26A03,
03E17

Retrieve articles in all journals with MSC (2010): 03E35, 26A03, 03E17

Additional Information

**Arnold W. Miller**

Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388

Email:
miller@math.wisc.edu

**Boaz Tsaban**

Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

Email:
tsaban@math.biu.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-10-10407-9

Received by editor(s):
October 21, 2009

Published electronically:
April 30, 2010

Communicated by:
Julia Knight

Article copyright:
© Copyright 2010
American Mathematical Society