On Possible Non-homeomorphic Substructures of the Real Line
نویسنده
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.
منابع مشابه
7. Ijpast-199-v6n1
We show a homeomorphic equivalent relation among the compact subspaces (Cantor middle-third sets, two-sided shift map of Cantor sets and ternary sets etc.) of the real line. Using such result, we show that there exists a chaotic homeomorphism of a compact subspace of the real line onto itself if and only if it is homeomorphic to the Cantor set.
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