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) a Jónsson cardinal; (ii) a sufficiently elementary submodel of the universe of sets with not homeomorphic to The reverse direction is a corollary to: Theorem 2. is Jónsson hereditarily separable, hereditarily Lindelöf, with . We further consider the large cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space. |