"initial topology" meaning in English

{
  "forms": [
    {
      "form": "initial topologies",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "initial topologies"
      },
      "expansion": "initial topology (plural initial 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"
        }
      ],
      "coordinate_terms": [
        {
          "word": "final topology"
        }
      ],
      "glosses": [
        "Given a set X, a family of topological spaces (M#x5F;i,T#x5F;i)#x5F;#x7B;i#x5C;inI#x7D; and a family of functions f#x5F;i#x3A;X#x5C;toM#x5F;i, the initial topology is the coarsest topology on X such that all f#x5F;i are continuous."
      ],
      "hyponyms": [
        {
          "word": "product topology"
        }
      ],
      "id": "en-initial_topology-en-noun-Ax0uF36C",
      "links": [
        [
          "topology",
          "topology"
        ],
        [
          "topological space",
          "topological space#English"
        ]
      ],
      "raw_glosses": [
        "(topology) Given a set X, a family of topological spaces (M#x5F;i,T#x5F;i)#x5F;#x7B;i#x5C;inI#x7D; and a family of functions f#x5F;i#x3A;X#x5C;toM#x5F;i, the initial topology is the coarsest topology on X such that all f#x5F;i are continuous."
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
      ]
    }
  ],
  "word": "initial topology"
}

{
  "coordinate_terms": [
    {
      "word": "final topology"
    }
  ],
  "forms": [
    {
      "form": "initial topologies",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "initial topologies"
      },
      "expansion": "initial topology (plural initial topologies)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "word": "product topology"
    }
  ],
  "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",
        "en:Topology"
      ],
      "glosses": [
        "Given a set X, a family of topological spaces (M#x5F;i,T#x5F;i)#x5F;#x7B;i#x5C;inI#x7D; and a family of functions f#x5F;i#x3A;X#x5C;toM#x5F;i, the initial topology is the coarsest topology on X such that all f#x5F;i are continuous."
      ],
      "links": [
        [
          "topology",
          "topology"
        ],
        [
          "topological space",
          "topological space#English"
        ]
      ],
      "raw_glosses": [
        "(topology) Given a set X, a family of topological spaces (M#x5F;i,T#x5F;i)#x5F;#x7B;i#x5C;inI#x7D; and a family of functions f#x5F;i#x3A;X#x5C;toM#x5F;i, the initial topology is the coarsest topology on X such that all f#x5F;i are continuous."
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
      ]
    }
  ],
  "word": "initial topology"
}