### 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 a exact consistency strength:
Theorem 1. The following are equiconsistent:
(i) ZFC + There Existskappa a Jonsson cardinal;
(ii) ZFC + There ExistsM a sufficiently elementary submodels of the universe of sets with R-M not homeomorphic to R. The reverse direction is a corollary to:
Theorem 2. cent is Jonsson double left right arrow There ExistsM <H (cent(+))There ExistsX(M) hereditarily separable, hereditarily Lindelof, T-3 with X not equal X-M.
We further consider the large cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space.

Translated title of the contribution | On possible non-homeomorphic substructures of the real line |
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Original language | English |

Pages (from-to) | 2771 - 2775 |

Number of pages | 5 |

Journal | Proceedings of the American Mathematical Society |

Volume | 130 (9) |

Publication status | Published - Sep 2002 |