On possible non-homeomorphic substructures of the real line

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.