{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-product-topology",
  "semantic_family_id": "sf-finite-product-topology",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限乘積拓撲"
  },
  "summary": {
    "zh-Hant": "有限多個拓撲空間的Cartesian product由open rectangles生成乘積拓撲。"
  },
  "statement": {
    "zh-Hant": "二因子finite models先驗每個factor topology，再建立完整Cartesian carrier與全部U×V basic rectangles，取任意聯集生成product topology。投影preimages必為open；對有限source Z，map z↦(f(z),g(z))連續恰當兩個coordinate maps連續。有限乘積與box topology相同，不能把這點外推到無限乘積；任意一族rectangles本身通常不是完整topology。"
  },
  "formula": {
    "tex": "P(X,Y)=t",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>P</mi><mo>(</mo><mi>X</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": "P",
          "glyph": "P"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "X",
            "glyph": "X"
          },
          {
            "type": "symbol",
            "name": "Y",
            "glyph": "Y"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "t",
        "glyph": "t"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "de29668a0b3cef48809c3625b20495f1920a79029b0147443e8eb034427f5caf"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "本節finite product限制至少一個因子；exact題族固定兩個非空factor，不主張零因子product convention。"
    },
    {
      "id": "a2",
      "zh-Hant": "兩個factor carrier有限且topology包含空集與全體並對任意聯集、有限交封閉。"
    },
    {
      "id": "a3",
      "zh-Hant": "ordered pairs、rectangles與set-of-sets均以extensional canonical form核對。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "finite_product_topology_operator",
      "role_zh": "有限乘積拓撲算子",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "first_topological_space",
      "role_zh": "第一拓撲空間",
      "scope": "representative_formula"
    },
    {
      "token": "Y",
      "role": "second_topological_space",
      "role_zh": "第二拓撲空間",
      "scope": "representative_formula"
    },
    {
      "token": "t",
      "role": "product_topology",
      "role_zh": "乘積拓撲",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-basis",
      "title_zh": "拓撲基底",
      "reason_zh": "open rectangles生成拓撲。"
    },
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "factor與product axioms。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "Cartesian carrier、聯集與交集。"
    },
    {
      "id": "mko-relation",
      "title_zh": "關係",
      "reason_zh": "ordered-pair carrier與coordinate maps。"
    },
    {
      "id": "mko-topological-continuity",
      "title_zh": "拓撲連續性",
      "reason_zh": "projections與coordinate criterion。"
    },
    {
      "id": "mko-composition",
      "title_zh": "函數合成",
      "reason_zh": "coordinate maps與universal property。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "factor topology與Cartesian carrier資格先核。",
      "rectangle basis須再對任意聯集封閉。",
      "projection與coordinate criterion只在完整生成拓撲上核對。"
    ],
    "common_misconception_zh": "把basic rectangles直接當全部opens，或因有限情形而宣稱無限product等於box topology。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "finite enumeration只證指定空間，不證一般product universal property。",
    "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": "加入任意乘積、box topology與Tychonoff邊界。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-product-topology.md 整理；2026-10-15 脈衝 R04（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"
  }
}