{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-quotient-topology",
  "semantic_family_id": "sf-quotient-topology",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "商拓撲"
  },
  "summary": {
    "zh-Hant": "滿射q:X→Y使U⊆Y恰在q逆像開時為開，從而賦予Y商拓撲。"
  },
  "statement": {
    "zh-Hant": "finite models先驗q為total surjection並完整列出fibers。定義tau_q={U⊆Y:q^{-1}(U)∈tau_X}，直接枚舉並驗拓撲公理、q的連續性與final-property：h:Y→Z連續當且僅當h∘q連續。只有saturated open subsets of X能作商開集的完整逆像；直接取open images一般錯誤。對任何total map（含非滿射），上述完整逆像的「當且僅當」條件仍唯一決定Y上的final topology；只有要求q連續則通常不唯一。未被命中的點不妨礙preimage-iff定義，但依此節點採用的慣例，非滿射q不稱為quotient map。"
  },
  "formula": {
    "tex": "Q(X,q,Y)=t",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>Q</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow><mo>=</mo><mi>t</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "Q",
          "glyph": "Q"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "X",
            "glyph": "X"
          },
          {
            "type": "symbol",
            "name": "q",
            "glyph": "q"
          },
          {
            "type": "symbol",
            "name": "Y",
            "glyph": "Y"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "t",
        "glyph": "t"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "beaa436749effba0c23925664cf690060ec981797fe2af549abf5e46f7d53a1c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "X、Y finite unique carriers，q為明示total surjection。"
    },
    {
      "id": "a2",
      "zh-Hant": "逆像、fiber、saturation與finite topology均完整枚舉。"
    }
  ],
  "symbols": [
    {
      "token": "Q",
      "role": "quotient_topology_operator",
      "role_zh": "商拓撲算子",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "source_topological_space",
      "role_zh": "來源拓撲空間",
      "scope": "representative_formula"
    },
    {
      "token": "q",
      "role": "surjective_quotient_map",
      "role_zh": "滿射商映射",
      "scope": "representative_formula"
    },
    {
      "token": "Y",
      "role": "quotient_carrier",
      "role_zh": "商carrier",
      "scope": "representative_formula"
    },
    {
      "token": "t",
      "role": "quotient_topology",
      "role_zh": "商拓撲",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "來源與商拓撲公理。"
    },
    {
      "id": "mko-quotient-set",
      "title_zh": "商集合",
      "reason_zh": "fibers與等價類。"
    },
    {
      "id": "mko-surjective",
      "title_zh": "滿射",
      "reason_zh": "quotient-map carrier責任。"
    },
    {
      "id": "mko-topological-continuity",
      "title_zh": "拓撲連續性",
      "reason_zh": "q與final property。"
    },
    {
      "id": "mko-composition",
      "title_zh": "函數合成",
      "reason_zh": "h與h∘q的final continuity criterion。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "作為此處的quotient map先核total surjection與fibers；一般total map的final topology仍由preimage-iff定義。",
      "opens由完整逆像條件定義，不是任意open image。",
      "finite final-property audit不取代一般證明。"
    ],
    "common_misconception_zh": "把q的open images直接當商拓撲，或把僅要求連續誤當成完整preimage-iff條件。非滿射仍可由preimage-iff唯一決定final topology，但依此處滿射慣例不稱quotient map。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "finite table只證指定quotient；Hausdorff等分離性不自動保留。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入identification spaces與quotient separation條件。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-quotient-topology.md 整理；2026-10-15 脈衝 R05（使用者手動接續 2026-11-17 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}