"Zariski topology" meaning in All languages combined

{
  "etymology_text": "After Russian-born American mathematician Oscar Zariski (1899—1986), regarded as one of the most influential algebraic geometers of the 20th century.",
  "forms": [
    {
      "form": "Zariski topologies",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Zariski topology (plural Zariski topologies)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Entries with translation boxes",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with German translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with Russian translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebraic geometry",
          "orig": "en:Algebraic geometry",
          "parents": [
            "Algebra",
            "Geometry",
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "The Zariski topology allows tools from topology to be used in the study of algebraic varieties, even when the underlying field is not a topological field.",
          "type": "example"
        },
        {
          "ref": "1994, V. I. Danilov, “II. Algebraic Varieties and Schemes”, in I. R. Shafarevich, editor, Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes, Springer, page 184:",
          "text": "Thus the Zariski open subsets are 'very big'; in particular, the Zariski topology is highly non-Hausdorff.\nA further difference with the classical topology is that the Zariski topology on the product X#x5C;timesY of two affine varieties is stronger than the product of the Zariski topologies on X and Y.",
          "type": "quote"
        },
        {
          "ref": "2008, Gert-Martin Greuel, Gerhard Pfister, A Singular Introduction to Commutative Algebra, 2nd edition, Springer, page 523:",
          "text": "In particular, “singular” is a local notion, where “local” so far was mainly considered with respect to the Zariski topology. However, since the Zariski topology is so coarse, small neighbourhoods in the Zariski topology might not be local enough.",
          "type": "quote"
        },
        {
          "ref": "2011, Gregor Kemper, A Course in Commutative Algebra, Springer, page 34:",
          "text": "If Y#x5C;subseteqKⁿ is an affine variety, then by definition the Zariski topology on Y has the subvarieties of Y as closed sets.",
          "type": "quote"
        }
      ],
      "glosses": [
        "Originally, a topology applicable to algebraic varieties, such that the closed sets are the variety's algebraic subvarieties; later, a generalisation in which the topological space is the set of prime ideals of a commutative ring and is called the spectrum of the ring."
      ],
      "id": "en-Zariski_topology-en-noun-eZbtlQKz",
      "links": [
        [
          "algebraic geometry",
          "algebraic geometry"
        ],
        [
          "topology",
          "topology"
        ],
        [
          "algebraic varieties",
          "algebraic variety"
        ],
        [
          "closed set",
          "closed set"
        ],
        [
          "algebraic subvarieties",
          "algebraic subvariety"
        ],
        [
          "topological space",
          "topological space"
        ],
        [
          "prime ideal",
          "prime ideal"
        ],
        [
          "commutative ring",
          "commutative ring"
        ],
        [
          "spectrum",
          "spectrum"
        ]
      ],
      "raw_glosses": [
        "(algebraic geometry) Originally, a topology applicable to algebraic varieties, such that the closed sets are the variety's algebraic subvarieties; later, a generalisation in which the topological space is the set of prime ideals of a commutative ring and is called the spectrum of the ring."
      ],
      "related": [
        {
          "english": "attributive form",
          "word": "Zariski"
        }
      ],
      "topics": [
        "algebraic-geometry",
        "geometry",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "de",
          "lang": "German",
          "sense": "Translations",
          "tags": [
            "feminine"
          ],
          "word": "Zariski-Topologie"
        },
        {
          "code": "ru",
          "lang": "Russian",
          "roman": "topologija Zarisskovo",
          "sense": "Translations",
          "tags": [
            "feminine"
          ],
          "word": "топология Зарисского"
        }
      ],
      "wikipedia": [
        "Oscar Zariski",
        "Zariski topology"
      ]
    }
  ],
  "word": "Zariski topology"
}

{
  "etymology_text": "After Russian-born American mathematician Oscar Zariski (1899—1986), regarded as one of the most influential algebraic geometers of the 20th century.",
  "forms": [
    {
      "form": "Zariski topologies",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Zariski topology (plural Zariski topologies)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "english": "attributive form",
      "word": "Zariski"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "English terms with usage examples",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "Terms with German translations",
        "Terms with Russian translations",
        "Translation table header lacks gloss",
        "en:Algebraic geometry"
      ],
      "examples": [
        {
          "text": "The Zariski topology allows tools from topology to be used in the study of algebraic varieties, even when the underlying field is not a topological field.",
          "type": "example"
        },
        {
          "ref": "1994, V. I. Danilov, “II. Algebraic Varieties and Schemes”, in I. R. Shafarevich, editor, Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes, Springer, page 184:",
          "text": "Thus the Zariski open subsets are 'very big'; in particular, the Zariski topology is highly non-Hausdorff.\nA further difference with the classical topology is that the Zariski topology on the product X#x5C;timesY of two affine varieties is stronger than the product of the Zariski topologies on X and Y.",
          "type": "quote"
        },
        {
          "ref": "2008, Gert-Martin Greuel, Gerhard Pfister, A Singular Introduction to Commutative Algebra, 2nd edition, Springer, page 523:",
          "text": "In particular, “singular” is a local notion, where “local” so far was mainly considered with respect to the Zariski topology. However, since the Zariski topology is so coarse, small neighbourhoods in the Zariski topology might not be local enough.",
          "type": "quote"
        },
        {
          "ref": "2011, Gregor Kemper, A Course in Commutative Algebra, Springer, page 34:",
          "text": "If Y#x5C;subseteqKⁿ is an affine variety, then by definition the Zariski topology on Y has the subvarieties of Y as closed sets.",
          "type": "quote"
        }
      ],
      "glosses": [
        "Originally, a topology applicable to algebraic varieties, such that the closed sets are the variety's algebraic subvarieties; later, a generalisation in which the topological space is the set of prime ideals of a commutative ring and is called the spectrum of the ring."
      ],
      "links": [
        [
          "algebraic geometry",
          "algebraic geometry"
        ],
        [
          "topology",
          "topology"
        ],
        [
          "algebraic varieties",
          "algebraic variety"
        ],
        [
          "closed set",
          "closed set"
        ],
        [
          "algebraic subvarieties",
          "algebraic subvariety"
        ],
        [
          "topological space",
          "topological space"
        ],
        [
          "prime ideal",
          "prime ideal"
        ],
        [
          "commutative ring",
          "commutative ring"
        ],
        [
          "spectrum",
          "spectrum"
        ]
      ],
      "raw_glosses": [
        "(algebraic geometry) Originally, a topology applicable to algebraic varieties, such that the closed sets are the variety's algebraic subvarieties; later, a generalisation in which the topological space is the set of prime ideals of a commutative ring and is called the spectrum of the ring."
      ],
      "topics": [
        "algebraic-geometry",
        "geometry",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "Oscar Zariski",
        "Zariski topology"
      ]
    }
  ],
  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "Translations",
      "tags": [
        "feminine"
      ],
      "word": "Zariski-Topologie"
    },
    {
      "code": "ru",
      "lang": "Russian",
      "roman": "topologija Zarisskovo",
      "sense": "Translations",
      "tags": [
        "feminine"
      ],
      "word": "топология Зарисского"
    }
  ],
  "word": "Zariski topology"
}