"sober space" meaning in English

{
  "forms": [
    {
      "form": "sober spaces",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "args": {},
      "expansion": "sober space (plural sober spaces)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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          "kind": "other",
          "name": "English entries with incorrect language header",
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          "source": "w"
        },
        {
          "kind": "other",
          "langcode": "en",
          "name": "Topological spaces",
          "orig": "en:Topological spaces",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Topology",
          "orig": "en:Topology",
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      "examples": [
        {
          "ref": "1975, Mathematica Scandinavica, Societates Mathematicae, page 318",
          "text": "For a reader more interested in function spaces than in functors, this concludes the description of content except to add that the classification of coadjoint G’s by sets B bearing a topological topology relativizes to T₀ spaces. For other readers: and trivially to sober spaces.",
          "type": "quotation"
        },
        {
          "text": "1983, Houston Journal of Mathematics, Volume 9, University of Houston, page 192,\nIn the Hofmann and Lawson paper, it is proved that the topological space Spec(L) is a locally quasicompact sober space […] ."
        },
        {
          "ref": "2002, P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, volume 2, Oxford University Press, page 492",
          "text": "Thus sober spaces are necessarily T₀.",
          "type": "quotation"
        },
        {
          "ref": "2003, A. Pultr, S. E. Rodabaugh, “Chapter 6: Lattice-Valued Frames, Functor Categories, And Classes of Sober Spaces”, in Stephen Ernest Rodabaugh, Erich Peter Klement, editors, Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Springer, page 155",
          "text": "How rich are sober and L-sober spaces, are there important examples? It is well-known [25] that Hausdorff spaces, and hence compact T₀ (and so finite T₀ spaces), are sober spaces in the traditional setting. Furthermore, the soberification of a space—the spectrum of the topology of a space—is always sober; and if the original space is not Hausdorff, then its soberification is a sober space which is not Hausdorff. So there are many non-Hausdorff sober spaces as well.",
          "type": "quotation"
        },
        {
          "ref": "2004, Ofer Gabber, “Notes on some t-structures”, in Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, François Loeser, editors, Geometric Aspects of Dwork Theory, volume 1, Walter de Gruyter, page 711",
          "text": "When X is a noetherian sober space the construction of loc. cit. can be extended to an arbitrary lower-semicontinuous function for the constructible topology.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A topological space of which every join-irreducible closed subset is the closure of exactly one point of the space."
      ],
      "hypernyms": [
        {
          "sense": "topological space",
          "word": "Kolmogorov space"
        },
        {
          "sense": "topological space",
          "word": "topological space"
        }
      ],
      "hyponyms": [
        {
          "sense": "topological space",
          "word": "Hausdorff space"
        }
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      "id": "en-sober_space-en-noun-H4cqBfkT",
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          "closed",
          "closed"
        ],
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          "closure",
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        ]
      ],
      "raw_glosses": [
        "(topology) A topological space of which every join-irreducible closed subset is the closure of exactly one point of the space."
      ],
      "related": [
        {
          "english": "latter form apparently more common",
          "word": "soberification"
        },
        {
          "english": "latter form apparently more common",
          "word": "sobrification"
        }
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
      ],
      "wikipedia": [
        "sober space"
      ]
    }
  ],
  "word": "sober space"
}

{
  "forms": [
    {
      "form": "sober spaces",
      "tags": [
        "plural"
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    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "sober space (plural sober spaces)",
      "name": "en-noun"
    }
  ],
  "hypernyms": [
    {
      "sense": "topological space",
      "word": "Kolmogorov space"
    },
    {
      "sense": "topological space",
      "word": "topological space"
    }
  ],
  "hyponyms": [
    {
      "sense": "topological space",
      "word": "Hausdorff space"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "english": "latter form apparently more common",
      "word": "soberification"
    },
    {
      "english": "latter form apparently more common",
      "word": "sobrification"
    }
  ],
  "senses": [
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        "en:Topological spaces",
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      "examples": [
        {
          "ref": "1975, Mathematica Scandinavica, Societates Mathematicae, page 318",
          "text": "For a reader more interested in function spaces than in functors, this concludes the description of content except to add that the classification of coadjoint G’s by sets B bearing a topological topology relativizes to T₀ spaces. For other readers: and trivially to sober spaces.",
          "type": "quotation"
        },
        {
          "text": "1983, Houston Journal of Mathematics, Volume 9, University of Houston, page 192,\nIn the Hofmann and Lawson paper, it is proved that the topological space Spec(L) is a locally quasicompact sober space […] ."
        },
        {
          "ref": "2002, P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, volume 2, Oxford University Press, page 492",
          "text": "Thus sober spaces are necessarily T₀.",
          "type": "quotation"
        },
        {
          "ref": "2003, A. Pultr, S. E. Rodabaugh, “Chapter 6: Lattice-Valued Frames, Functor Categories, And Classes of Sober Spaces”, in Stephen Ernest Rodabaugh, Erich Peter Klement, editors, Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Springer, page 155",
          "text": "How rich are sober and L-sober spaces, are there important examples? It is well-known [25] that Hausdorff spaces, and hence compact T₀ (and so finite T₀ spaces), are sober spaces in the traditional setting. Furthermore, the soberification of a space—the spectrum of the topology of a space—is always sober; and if the original space is not Hausdorff, then its soberification is a sober space which is not Hausdorff. So there are many non-Hausdorff sober spaces as well.",
          "type": "quotation"
        },
        {
          "ref": "2004, Ofer Gabber, “Notes on some t-structures”, in Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, François Loeser, editors, Geometric Aspects of Dwork Theory, volume 1, Walter de Gruyter, page 711",
          "text": "When X is a noetherian sober space the construction of loc. cit. can be extended to an arbitrary lower-semicontinuous function for the constructible topology.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A topological space of which every join-irreducible closed subset is the closure of exactly one point of the space."
      ],
      "links": [
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          "topological space",
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          "irreducible",
          "irreducible"
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        [
          "closed",
          "closed"
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          "closure",
          "closure"
        ]
      ],
      "raw_glosses": [
        "(topology) A topological space of which every join-irreducible closed subset is the closure of exactly one point of the space."
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
      ],
      "wikipedia": [
        "sober space"
      ]
    }
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  "word": "sober space"
}