{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-topological-basis",
  "semantic_family_id": "sf-topological-basis",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "拓撲基底"
  },
  "summary": {
    "zh-Hant": "拓撲基底以覆蓋carrier與逐點局部交細化兩條公理生成拓撲。"
  },
  "statement": {
    "zh-Hant": "每個basis element是X的子集且全部聯集覆蓋X；對x∈B1∩B2，存在B3使x∈B3⊆B1∩B2。若聲稱為既有拓撲的basis，還需basis elements皆開且每個open set為basis聯集。basis不可與subbasis混同。"
  },
  "formula": {
    "tex": "B(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "B",
        "glyph": "B"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "91b4e6f5f35d86b528823e5c2357eabc054277f2e5df5a824db12c61e01bb3bf"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限carrier與完整basis candidate。"
    },
    {
      "id": "a2",
      "zh-Hant": "既有拓撲題列出全部open sets並精確驗拓撲公理。"
    }
  ],
  "symbols": [
    {
      "token": "B",
      "role": "topological_basis",
      "role_zh": "拓撲基底",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "carrier",
      "role_zh": "底集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "生成或比較的拓撲。"
    },
    {
      "id": "mko-open-closed-set",
      "title_zh": "開集與閉集",
      "reason_zh": "basis elements與open sets。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "覆蓋、交與聯。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "覆蓋與局部交細化缺一不可。",
      "生成拓撲是所有basis elements聯集。",
      "相對既有拓撲還需雙向生成責任。"
    ],
    "common_misconception_zh": "任何覆蓋族都是basis，或把subbasis直接當basis。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "exact enumeration只證明宣告的有限carrier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subspace、product與quotient topologies。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-topological-basis.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-11-07 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"
  }
}