"disjoint union topology" meaning in English

{
  "forms": [
    {
      "form": "disjoint union topologies",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "disjoint union topology (plural disjoint union 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": "topical",
          "langcode": "en",
          "name": "Topology",
          "orig": "en:Topology",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1978, Maria Louise Shea Terrell, Topological 2-categories and Principal Topological Categories, University of Virginia, page 19",
          "text": "The reader will recall that the morphism and object sets of A'#x3D;#x5C;coprod#x5F;#x7B;p,q#x5C;inP#x7D;A#x5F;c(q,p) were not necessarily given the disjoint union topologies.",
          "type": "quotation"
        },
        {
          "ref": "2003, John M. Lee, Introduction to Smooth Manifolds, Springer, page 548",
          "text": "Given any indexed collection of topological spaces #x5C;left#x5C;#x7B;X#x5F;#x5C;alpha#x5C;right#x5C;#x7D;#x5F;#x7B;#x5C;alpha#x5C;inA#x7D;, we define the disjoint union topology on #x5C;textstyle#x5C;coprod#x5F;#x7B;#x5C;alpha#x5C;inA#x7D;X#x5F;#x5C;alpha by declaring a subset of #x5C;textstyle#x5C;coprod#x5F;#x7B;#x5C;alpha#x5C;inA#x7D;X#x5F;#x5C;alpha to be open if and only if its intersection with each X#x5F;#x5C;alpha is open in X#x5F;#x5C;alpha.",
          "type": "quotation"
        },
        {
          "ref": "2005, Friedrich Ischebeck, Ravi A. Rao, Ideals and Reality: Projective Modules and Number of Generators of Ideals, Springer, page 118",
          "text": "And the set Gⁿ(#x5C;mathbb#x7B;F#x7D;) of all sub vector spaces of #x5C;mathbb#x7B;F#x7D;ⁿ, i.e. the disjoint union of the Gⁿ#x5F;k(#x5C;mathbb#x7B;F#x7D;), gets the disjoint union topology.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A topology which is applicable to the disjoint union of a given set of topological spaces and is the largest (most inclusive) topology that preserves the continuity of each contributing space as represented in the union."
      ],
      "id": "en-disjoint_union_topology-en-noun--REpFjra",
      "links": [
        [
          "topology",
          "topology"
        ],
        [
          "disjoint union",
          "disjoint union"
        ],
        [
          "topological space",
          "topological space"
        ]
      ],
      "raw_glosses": [
        "(topology) A topology which is applicable to the disjoint union of a given set of topological spaces and is the largest (most inclusive) topology that preserves the continuity of each contributing space as represented in the union."
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
      ]
    }
  ],
  "word": "disjoint union topology"
}

{
  "forms": [
    {
      "form": "disjoint union topologies",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "disjoint union topology (plural disjoint union topologies)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Topology"
      ],
      "examples": [
        {
          "ref": "1978, Maria Louise Shea Terrell, Topological 2-categories and Principal Topological Categories, University of Virginia, page 19",
          "text": "The reader will recall that the morphism and object sets of A'#x3D;#x5C;coprod#x5F;#x7B;p,q#x5C;inP#x7D;A#x5F;c(q,p) were not necessarily given the disjoint union topologies.",
          "type": "quotation"
        },
        {
          "ref": "2003, John M. Lee, Introduction to Smooth Manifolds, Springer, page 548",
          "text": "Given any indexed collection of topological spaces #x5C;left#x5C;#x7B;X#x5F;#x5C;alpha#x5C;right#x5C;#x7D;#x5F;#x7B;#x5C;alpha#x5C;inA#x7D;, we define the disjoint union topology on #x5C;textstyle#x5C;coprod#x5F;#x7B;#x5C;alpha#x5C;inA#x7D;X#x5F;#x5C;alpha by declaring a subset of #x5C;textstyle#x5C;coprod#x5F;#x7B;#x5C;alpha#x5C;inA#x7D;X#x5F;#x5C;alpha to be open if and only if its intersection with each X#x5F;#x5C;alpha is open in X#x5F;#x5C;alpha.",
          "type": "quotation"
        },
        {
          "ref": "2005, Friedrich Ischebeck, Ravi A. Rao, Ideals and Reality: Projective Modules and Number of Generators of Ideals, Springer, page 118",
          "text": "And the set Gⁿ(#x5C;mathbb#x7B;F#x7D;) of all sub vector spaces of #x5C;mathbb#x7B;F#x7D;ⁿ, i.e. the disjoint union of the Gⁿ#x5F;k(#x5C;mathbb#x7B;F#x7D;), gets the disjoint union topology.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A topology which is applicable to the disjoint union of a given set of topological spaces and is the largest (most inclusive) topology that preserves the continuity of each contributing space as represented in the union."
      ],
      "links": [
        [
          "topology",
          "topology"
        ],
        [
          "disjoint union",
          "disjoint union"
        ],
        [
          "topological space",
          "topological space"
        ]
      ],
      "raw_glosses": [
        "(topology) A topology which is applicable to the disjoint union of a given set of topological spaces and is the largest (most inclusive) topology that preserves the continuity of each contributing space as represented in the union."
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
      ]
    }
  ],
  "word": "disjoint union topology"
}