On possible non-homeomorphic substructures of the real line
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Abstract:
We consider the problem, raised by Kunen and Tall, of whether the real continuum can have non-homeomorphic versions in different submodels of the universe of all sets. This requires large cardinals, and we obtain an exact consistency strength:
Theorem 1. The following are equiconsistent:
(i) $ZFC + \exists \kappa$ a Jónsson cardinal;
(ii) $ZFC + \exists M$ a sufficiently elementary submodel of the universe of sets with ${\mathbb R}_M$ not homeomorphic to ${\mathbb R}.$ The reverse direction is a corollary to:
Theorem 2. $\mathfrak {c}$ is Jónsson $\Longleftrightarrow \exists M \prec H(\mathfrak {c}^+)\exists X_M$ hereditarily separable, hereditarily Lindelöf, $T_3$ with $X \neq X_M$. We further consider the large cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space.
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Additional Information
- P. D. Welch
- Affiliation: Department of Mathematics, University of Bristol, Bristol BS8 1TW, England – and – Department Institut für Formale Logik, Währingerstr 25, A-1090 Wien, Austria
- Address at time of publication: Mathematisches Institut, Beringstrasse 6, Bonn, D-53115, Germany
- Email: welch@logic.univie.ac.at
- Received by editor(s): January 16, 2001
- Received by editor(s) in revised form: March 27, 2001
- Published electronically: February 12, 2002
- Communicated by: Carl G. Jockusch, Jr.
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2771-2775
- MSC (2000): Primary 54A05, 03E35, 03E02, 54A35, 03E55; Secondary 54B05
- DOI: https://doi.org/10.1090/S0002-9939-02-06385-2
- MathSciNet review: 1900884