On possible non-homeomorphic substructures of the real line

P. D. Welch
2002 Proceedings of the American Mathematical Society  
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) ZF C + ∃κ a Jónsson cardinal ; (ii) ZF C + ∃M a sufficiently elementary submodel of the universe of sets with R M not homeomorphic to R. The reverse direction is a corollary to: We further consider the
more » ... cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space.
doi:10.1090/s0002-9939-02-06385-2 fatcat:qf4jie2dujbizoxe4hklhgg2by