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Transactions of the American Mathematical Society

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Derived topologies on ordinals and stationary reflection

Author: Joan Bagaria
Journal: Trans. Amer. Math. Soc. 371 (2019), 1981-2002
MSC (2010): Primary 03B45, 03E35, 03E55, 03F45, 54G12; Secondary 03E10, 03E45, 03E05, 54A35
Published electronically: September 20, 2018
MathSciNet review: 3894041
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Abstract: We study the transfinite sequence of topologies on the ordinal numbers that is obtained through successive closure under Cantor's derivative operator on sets of ordinals, starting from the usual interval topology. We characterize the non-isolated points in the $ \xi $th topology as those ordinals that satisfy a strong iterated form of stationary reflection, which we call $ \xi $-simultaneous-reflection. We prove some properties of the ideals of non-$ \xi $-simultaneous-stationary sets and identify their tight connection with indescribable cardinals. We then introduce a new natural notion of $ \Pi ^1_\xi $-indescribability, for any ordinal $ \xi $, which extends to the transfinite the usual notion of $ \Pi ^1_n$-indescribability, and prove that in the constructible universe $ L$, a regular cardinal is $ (\xi +1)$-simultaneously-reflecting if and only if it is $ \Pi ^1_\xi $-indescribable, a result that generalizes to all ordinals $ \xi $ previous results of Jensen [Ann. Math. Logic 4 (1972), pp. 229-308] in the case $ \xi =2$, and Bagaria-Magidor-Sakai [Israel J. Math. 208 (2015), pp. 1-11] in the case $ \xi =n$. This yields a complete characterization in $ L$ of the non-discreteness of the $ \xi $-topologies, both in terms of iterated stationary reflection and in terms of indescribability.

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Joan Bagaria
Affiliation: ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Catalonia, Spain – and – Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Catalonia, Spain

Keywords: Derived topology, ordinals, stationary reflection, indescribable cardinals
Received by editor(s): July 8, 2016
Received by editor(s) in revised form: August 6, 2017
Published electronically: September 20, 2018
Additional Notes: Part of this research was done while the author was a Simons Foundation Fellow at Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, in the programme “Mathematical, Foundational and Computational Aspects of the Higher Infinite” (HIF)", September-December 2015, supported by EPSRC Grant Number EP/K032208/1. The research work was also partially supported by the Spanish Government under grant MTM2014-59178-P, and by the Generalitat de Catalunya (Catalan Government) under grant SGR 437-2014.
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