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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On possible non-homeomorphic substructures of the real line


Author: P. D. Welch
Journal: Proc. Amer. Math. Soc. 130 (2002), 2771-2775
MSC (2000): Primary 54A05, 03E35, 03E02, 54A35, 03E55; Secondary 54B05
Published electronically: February 12, 2002
MathSciNet review: 1900884
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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

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06385-2
PII: S 0002-9939(02)06385-2
Keywords: Real continuum, subspaces, J\'{o}nsson cardinals
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.
Article copyright: © Copyright 2002 American Mathematical Society