Hausdorff gap

Named after German mathematician Felix Hausdorff (1868–1942), who published proof of the first example in 1909.

Hausdorff gap (plural Hausdorff gaps)

1. (set theory, order theory) A pair of collections of integer sequences such that there is no integer sequence lying between the two.

   The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.^[Wikipedia]

   • 1995, Robert M. Solovay, “*1970 [Godel's ontological proof]”, in Kurt Gödel, edited by Solomon Feferman, John W. Dawson Jr., Warren Goldfarb, Charles Parsons, and Robert M. Solovay, Kurt Gödel: Collected Works: Volume III, Oxford University Press, page 419:

     Of course, Hausdorff did not talk of models of set theory or prove absoluteness results. He gave a direct construction in ZFC of what is now called a Hausdorff gap and proved properties 1 through 4 for his construction. The notion of a Hausdorff gap and the proof that Hausdorff gaps are indestructible under cardinal preserving extensions (property 5 above) are due to Kunen (unpublished).

   • 1997, Winfried Just, Martin Weese, Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 118:

     Let ${\displaystyle \langle \langle a_{\epsilon }:\epsilon <\omega _{1}\rangle \langle b_{\epsilon }:\epsilon <\omega _{1}\rangle \rangle }$ be a Hausdorff gap.

   • 2013, Ilijas Farah, Eric Wofsey, 3: Set theory and operator algebras, James Cummings, Ernest Schimmerling (editors), Appalachian Set Theory: 2006-2012, London Mathematical Society, Cambridge University Press, page 101,

     This family is one of the instances of incompactness of ${\displaystyle \omega _{1}}$ that are provable in ZFC, along with Hausdorff gaps, special Aronszajn trees, or nontrivial coherent families of partial functions.

• Rothberger gap

• Hausdorff gap on Encyclopedia of Mathematics