{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-topological-connected-component",
  "semantic_family_id": "sf-topological-connected-component",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "拓撲連通分支"
  },
  "summary": {
    "zh-Hant": "指定點的連通分支是包含該點的極大連通子集；所有分支分割空間且各自閉。"
  },
  "statement": {
    "zh-Hant": "finite topology families完整枚舉subspace topologies與所有connected subsets；含x的connected subsets之union給唯一inclusion-maximal component。再核components兩兩相等或不交、覆蓋X並在指定finite model中closed。有限拓撲因只有有限多components而使每個component同時open，這是有限特例；一般空間的component不必open，Q的通常subspace topology中singletons是components但不open。不得把最大基數、finite-graph connected component或其他結構混同。"
  },
  "formula": {
    "tex": "K(X,x)=C",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>K</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>C</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "K",
          "glyph": "K"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "X",
            "glyph": "X"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f3963cae4140c47ea830b8d4132fe11f131686392fb0357d33b6b0f647312a6c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "finite families使用完整carrier與完整topology，不以鄰接圖代替opens。"
    },
    {
      "id": "a2",
      "zh-Hant": "connected subset以subspace topology不存在非平凡separation判定。"
    }
  ],
  "symbols": [
    {
      "token": "K",
      "role": "connected_component_operator",
      "role_zh": "連通分支算子",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "topological_space",
      "role_zh": "拓撲空間",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "base_point",
      "role_zh": "指定點",
      "scope": "representative_formula"
    },
    {
      "token": "C",
      "role": "maximal_connected_subset",
      "role_zh": "極大連通子集",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-connected-topological-space",
      "title_zh": "連通拓撲空間",
      "reason_zh": "connected subset語義。"
    },
    {
      "id": "mko-equivalence-class",
      "title_zh": "等價類",
      "reason_zh": "component partition。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "極大connected subset。"
    },
    {
      "id": "mko-closure-and-interior",
      "title_zh": "閉包與內部",
      "reason_zh": "一般component closed性。"
    },
    {
      "id": "mko-connected-component",
      "title_zh": "圖的連通分量",
      "reason_zh": "同名異結構的精確區分。"
    },
    {
      "id": "mko-subspace-topology",
      "title_zh": "子空間拓撲",
      "reason_zh": "subset connectedness與Q boundary。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整subspace connectedness枚舉先於inclusion-maximality，不能改用最大基數。",
      "component partition與closed性逐項核對。",
      "finite-space component為clopen是有限特例；Q的symbolic boundary保護一般不open。"
    ],
    "common_misconception_zh": "把最大基數connected subset、graph連通分量或有限空間中open component的現象當成一般定義。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "finite enumeration只證指定topology；一般maximality與closed性仍需證明。",
    "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quasicomponents與local connectedness。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-topological-connected-component.md 整理；2026-10-15 脈衝 R07（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"
  }
}