On Possible Non-homeomorphic Substructures of the Real Line

نویسنده

  • P. D. WELCH
چکیده

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 + ∃κ a Jónsson cardinal ; (ii) ZFC + ∃M a sufficiently elementary submodel of the universe of sets with RM not homeomorphic to R. The reverse direction is a corollary to: Theorem 2. c is Jónsson ⇐⇒ ∃M ≺ H(c)∃XM hereditarily separable, hereditarily Lindelöf, T3 with X 6= XM . We further consider the large cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space.

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تاریخ انتشار 2002