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 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.