A topological reflection principle equivalent to Shelah’s strong hypothesis
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- by Assaf Rinot
- Proc. Amer. Math. Soc. 136 (2008), 4413-4416
- DOI: https://doi.org/10.1090/S0002-9939-08-09411-2
- Published electronically: July 1, 2008
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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
- R. Hodel, Cardinal functions. I, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1–61. MR 776620
- R. E. Hodel and J. E. Vaughan, Reflection theorems for cardinal functions, Topology Appl. 100 (2000), no. 1, 47–66. Special issue in honor of Howard H. Wicke. MR 1731704, DOI 10.1016/S0166-8641(99)00056-5
- M. Ismail and A. Szymanski, A topological equivalence of the singular cardinals hypothesis, Proc. Amer. Math. Soc. 123 (1995), no. 3, 971–973. MR 1285997, DOI 10.1090/S0002-9939-1995-1285997-0
- S. Mrówka, On completely regular spaces, Fund. Math. 41 (1954), 105–106. MR 63650, DOI 10.4064/fm-41-1-105-106
- Assaf Rinot, On the consistency strength of the Milner-Sauer conjecture, Ann. Pure Appl. Logic 140 (2006), no. 1-3, 110–119. MR 2224053, DOI 10.1016/j.apal.2005.09.012
- Assaf Rinot, On topological spaces of singular density and minimal weight, Topology Appl. 155 (2007), no. 3, 135–140. MR 2370368, DOI 10.1016/j.topol.2007.09.013
- Saharon Shelah, Cardinal arithmetic for skeptics, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 197–210. MR 1112424, DOI 10.1090/S0273-0979-1992-00261-6
- Saharon Shelah, Advances in cardinal arithmetic, Finite and infinite combinatorics in sets and logic (Banff, AB, 1991) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 411, Kluwer Acad. Publ., Dordrecht, 1993, pp. 355–383. MR 1261217
- Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1318912
Bibliographic Information
- Assaf Rinot
- Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
- MR Author ID: 785097
- Email: assaf@rinot.com
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2431057