{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-open-closed-set",
  "semantic_family_id": "sf-open-closed-set",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "開集與閉集"
  },
  "summary": {
    "zh-Hant": "在固定拓撲空間中，開集屬於拓撲；閉集的補集為開集，同一集合可以同時開且閉。"
  },
  "statement": {
    "zh-Hant": "給定拓撲空間 (X,τ)，U⊆X 稱為開集當且僅當 U∈τ；C⊆X 稱為閉集當且僅當 X\\C∈τ。開與閉不是互斥分類，∅ 與 X 永遠同時開且閉。"
  },
  "formula": {
    "tex": "C\\setminus U",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>∖</mo><mi>U</mi></mrow></math>",
    "semantic_ast": {
      "type": "set_difference",
      "left": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "right": {
        "type": "symbol",
        "name": "U",
        "glyph": "U"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f24db1adc25ff4487d972d9be2e30c0333c1d7a4fbc2b9b3223a25b04005bc79"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "底集合 X 與其拓撲 τ 已固定。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有補集都相對於同一底集合 X；候選集合先確認為 X 的子集合。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "closed_candidate",
      "role_zh": "候選閉集",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "underlying_set",
      "role_zh": "底集合",
      "scope": "representative_formula"
    },
    {
      "token": "U",
      "role": "open_candidate",
      "role_zh": "候選開集",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "開集族由固定拓撲提供。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "開閉候選都是底集合的子集合。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "閉集定義使用相對補集。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "閉集不是『不是開集』；集合可以只開、只閉、同時開閉，或兩者皆非。",
      "任意開集聯集與有限開集交仍開；對偶地，任意閉集交與有限閉集聯集仍閉。有限模型可以完整列出開閉與 clopen 集。",
      "補集對偶必須固定同一底集合。若偷偷改變 universe，開閉判定失去原拓撲語境。"
    ],
    "common_misconception_zh": "把 closed 解讀成 not open，或忘記補集相對的底集合。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限拓撲可完整枚舉開閉集；這不取代一般任意交／聯公理的形式證明。",
    "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": "綁定 Mathlib IsOpen、IsClosed、complement 與有限／任意閉集封閉定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-open-closed-set.md 整理；2026-10-13 脈衝 R01（使用者手動接續 2026-10-19 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}