Archive for Mathematical Logic 53 (1-2):57-64 (2014)

Abstract
Todorčević (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show that ${\mathsf{AD}_\mathbb{R}}$ for pointclasses Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$
Keywords Hausdorff gaps  Descriptive set theory  Infinite games
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DOI 10.1007/s00153-013-0355-6
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Infinite Combinatorics and Definability.Arnold W. Miller - 1989 - Annals of Pure and Applied Logic 41 (2):179-203.

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