"Hausdorff gap" meaning in English

{
  "etymology_text": "Named after German mathematician Felix Hausdorff (1868–1942), who published proof of the first example in 1909.",
  "forms": [
    {
      "form": "Hausdorff gaps",
      "tags": [
        "plural"
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  ],
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      "expansion": "Hausdorff gap (plural Hausdorff gaps)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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          "name": "Set theory",
          "orig": "en:Set theory",
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      "examples": [
        {
          "text": "The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.",
          "type": "example"
        },
        {
          "ref": "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",
          "text": "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proof that Hausdorff gaps are indestructible under cardinal preserving extensions (property 5 above) are due to Kunen (unpublished).",
          "type": "quotation"
        },
        {
          "ref": "1997, Winfried Just, Martin Weese, Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 118",
          "text": "Let #x5C;langle#x5C;langlea#x5F;#x5C;epsilon#x3A;#x5C;epsilon#x3C;#x5C;omega#x5F;1#x5C;rangle#x5C;langleb#x5F;#x5C;epsilon#x3A;#x5C;epsilon#x3C;#x5C;omega#x5F;1#x5C;rangle#x5C;rangle be a Hausdorff gap.",
          "type": "quotation"
        },
        {
          "text": "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,\nThis family is one of the instances of incompactness of ω₁ that are provable in ZFC, along with Hausdorff gaps, special Aronszajn trees, or nontrivial coherent families of partial functions."
        }
      ],
      "glosses": [
        "A pair of collections of integer sequences such that there is no integer sequence lying between the two."
      ],
      "id": "en-Hausdorff_gap-en-noun-ojixX6c8",
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          "pair",
          "pair"
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        [
          "integer",
          "integer"
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        [
          "sequence",
          "sequence"
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      ],
      "raw_glosses": [
        "(set theory, order theory) A pair of collections of integer sequences such that there is no integer sequence lying between the two."
      ],
      "related": [
        {
          "word": "Rothberger gap"
        }
      ],
      "topics": [
        "mathematics",
        "order-theory",
        "sciences",
        "set-theory"
      ],
      "wikipedia": [
        "Felix Hausdorff",
        "Hausdorff gap"
      ]
    }
  ],
  "word": "Hausdorff gap"
}

{
  "etymology_text": "Named after German mathematician Felix Hausdorff (1868–1942), who published proof of the first example in 1909.",
  "forms": [
    {
      "form": "Hausdorff gaps",
      "tags": [
        "plural"
      ]
    }
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  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Rothberger gap"
    }
  ],
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      "examples": [
        {
          "text": "The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.",
          "type": "example"
        },
        {
          "ref": "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",
          "text": "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).",
          "type": "quotation"
        },
        {
          "ref": "1997, Winfried Just, Martin Weese, Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 118",
          "text": "Let #x5C;langle#x5C;langlea#x5F;#x5C;epsilon#x3A;#x5C;epsilon#x3C;#x5C;omega#x5F;1#x5C;rangle#x5C;langleb#x5F;#x5C;epsilon#x3A;#x5C;epsilon#x3C;#x5C;omega#x5F;1#x5C;rangle#x5C;rangle be a Hausdorff gap.",
          "type": "quotation"
        },
        {
          "text": "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,\nThis family is one of the instances of incompactness of ω₁ that are provable in ZFC, along with Hausdorff gaps, special Aronszajn trees, or nontrivial coherent families of partial functions."
        }
      ],
      "glosses": [
        "A pair of collections of integer sequences such that there is no integer sequence lying between the two."
      ],
      "links": [
        [
          "set theory",
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        ],
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          "pair",
          "pair"
        ],
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          "integer"
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        [
          "sequence",
          "sequence"
        ]
      ],
      "raw_glosses": [
        "(set theory, order theory) A pair of collections of integer sequences such that there is no integer sequence lying between the two."
      ],
      "topics": [
        "mathematics",
        "order-theory",
        "sciences",
        "set-theory"
      ],
      "wikipedia": [
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      ]
    }
  ],
  "word": "Hausdorff gap"
}