{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-topological-boundary",
  "semantic_family_id": "sf-topological-boundary",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "拓撲邊界"
  },
  "summary": {
    "zh-Hant": "集合A的拓撲邊界為closure(A)減去interior(A)，亦等於closure(A)交closure(complement A)。"
  },
  "statement": {
    "zh-Hant": "完整驗有限拓撲後精確計算closure與interior。保留boundary(A)=boundary(complement A)、空集與全空間邊界，以及拓撲相對性。neighborhood contact若使用必須枚舉全部open neighborhoods。"
  },
  "formula": {
    "tex": "B(A)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>B</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "B",
        "glyph": "B"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "A",
          "glyph": "A"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "a01bbeb0844246132ad4fa7fb458128281305e1bdfd9fcbc39f45fe301a2b0d4"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限載體、拓撲與A。"
    },
    {
      "id": "a2",
      "zh-Hant": "補集相對同一載體；未供度量時不得使用metric boundary。"
    }
  ],
  "symbols": [
    {
      "token": "B",
      "role": "topological_boundary",
      "role_zh": "拓撲邊界",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "subset",
      "role_zh": "指定子集",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-closure-and-interior",
      "title_zh": "閉包與內部",
      "reason_zh": "boundary=closure減interior。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "差集、補集與交集。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "由完整閉包與內部計算。",
      "A與補集有相同boundary。",
      "空集與全空間的boundary皆空；結果依拓撲。"
    ],
    "common_misconception_zh": "把metric圖形外觀當成未指定拓撲的boundary。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限完整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connectedness與分離性。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-topological-boundary.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-11-02 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"
  }
}