{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-closure-and-interior",
  "semantic_family_id": "sf-closure-and-interior",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "拓撲閉包與內部"
  },
  "summary": {
    "zh-Hant": "集合閉包是所有閉超集的交，內部是所有開子集的聯；兩者相對於指定拓撲。"
  },
  "statement": {
    "zh-Hant": "題目先完整驗證有限拓撲，再由全部閉超集或開子集精確求閉包與內部。空集、全空間及拓撲相對性均保留，不暗中使用度量。"
  },
  "formula": {
    "tex": "C(A)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "A",
          "glyph": "A"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "0c5a166ffa7943d44aa027a123c6c02dba6632b46a623d2ec8a27b19ebd5d0d6"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "有限載體與完整canonical拓撲集合族。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有集合皆為載體子集，補集相對同一載體。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "closure_interior_record",
      "role_zh": "閉包／內部紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "subset",
      "role_zh": "指定子集",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-open-closed-set",
      "title_zh": "開集與閉集",
      "reason_zh": "閉超集與開子集來源。"
    },
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "固定完整拓撲。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "交、聯與補集。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "閉包取全部閉超集之交。",
      "內部取全部開子集之聯。",
      "結果依拓撲改變，不自動是度量球概念。"
    ],
    "common_misconception_zh": "只找一個閉超集／開子集，或未驗拓撲就計算。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "完整有限枚舉只證指定拓撲實例。",
    "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": "加入邊界、連通與緊緻。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-closure-and-interior.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-01 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}