Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A topological reflection principle equivalent to Shelah's strong hypothesis


Author: Assaf Rinot
Journal: Proc. Amer. Math. Soc. 136 (2008), 4413-4416
MSC (2000): Primary 03E04; Secondary 54G15, 03E65
DOI: https://doi.org/10.1090/S0002-9939-08-09411-2
Published electronically: July 1, 2008
MathSciNet review: 2431057
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We notice that Shelah's Strong Hypothesis is equivalent to the following reflection principle:

Suppose $ \langle X,\tau\rangle$ is a first-countable space whose density is a regular cardinal, $ \kappa$. If every separable subspace of $ X$ is of cardinality at most $ \kappa$, then the cardinality of $ X$ is $ \kappa$.


References [Enhancements On Off] (What's this?)

  • 1. R. E. Hodel.
    Cardinal functions. I.
    In Handbook of set-theoretic topology, pages 1-61. North-Holland, Amsterdam, 1984. MR 776620 (86j:54007)
  • 2. R. E. Hodel and J. E. Vaughan.
    Reflection theorems for cardinal functions.
    Topology Appl., 100(1):47-66, 2000.
    Special issue in honor of Howard H. Wicke. MR 1731704 (2001b:54004)
  • 3. M. Ismail and A. Szymanski.
    A topological equivalence of the singular cardinals hypothesis.
    Proc. Amer. Math. Soc., 123(3):971-973, 1995. MR 1285997 (95k:03076)
  • 4. S. Mrówka.
    On completely regular spaces.
    Fund. Math., 41:105-106, 1954. MR 0063650 (16:157b)
  • 5. A. Rinot.
    On the consistency strength of the Milner-Sauer conjecture.
    Ann. Pure Appl. Logic, 140(1-3):110-119, 2006. MR 2224053 (2007a:03067)
  • 6. A. Rinot.
    On topological spaces of singular density and minimal weight.
    Topology Appl., 155(3):135-140, 2007. MR 2370368
  • 7. S. Shelah.
    Cardinal arithmetic for skeptics.
    Bull. Amer. Math. Soc. (N.S.), 26(2):197-210, 1992. MR 1112424 (92h:03071)
  • 8. S. Shelah.
    Advances in cardinal arithmetic.
    In Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), volume 411 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 355-383. Kluwer Acad. Publ., Dordrecht, 1993. MR 1261217 (95h:03112)
  • 9. S. Shelah.
    $ \aleph_{\omega+1}$ has a Jonsson algebra.
    In Cardinal Arithmetic, pages 34-116. Oxford Logic Guides 29, Oxford Univ. Press, 1994. MR 1318912 (96e:03001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03E04, 54G15, 03E65

Retrieve articles in all journals with MSC (2000): 03E04, 54G15, 03E65


Additional Information

Assaf Rinot
Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Email: assaf@rinot.com

DOI: https://doi.org/10.1090/S0002-9939-08-09411-2
Keywords: Shelah's Strong Hypothesis, first countable, countably tight.
Received by editor(s): September 28, 2007
Received by editor(s) in revised form: November 3, 2007
Published electronically: July 1, 2008
Additional Notes: The author would like to thank his Ph.D. advisor, M. Gitik, for his comments and remarks.
Communicated by: Julia Knight
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society