{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-hausdorff-space",
  "semantic_family_id": "sf-hausdorff-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Hausdorff空間"
  },
  "summary": {
    "zh-Hant": "拓撲空間X若每對不同點都有不交open neighborhoods即為Hausdorff。"
  },
  "statement": {
    "zh-Hant": "必須forall x≠y存在open U,V，使x∈U、y∈V且U∩V=empty。一個失敗pair即可證non-Hausdorff；有限sample成功不足以證一般空間。本輪完整有限topology exhaustive驗證。empty與singleton vacuously Hausdorff；finite Hausdorff implies discrete不可外推至infinite spaces。"
  },
  "formula": {
    "tex": "H(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "H",
        "glyph": "H"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ada1c5766d80d2521544fa32742789460620c60846402eed28efc53c807ddce2"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "carrier finite、labels unique且topology完整canonical。"
    },
    {
      "id": "a2",
      "zh-Hant": "pair witnesses必須實際屬topology並包含相應points。"
    }
  ],
  "symbols": [
    {
      "token": "H",
      "role": "Hausdorff_predicate",
      "role_zh": "Hausdorff述詞",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "topological_space",
      "role_zh": "拓撲空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "open sets。"
    },
    {
      "id": "mko-neighborhood",
      "title_zh": "鄰域",
      "reason_zh": "point neighborhoods。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "forall distinct pairs。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "different points require disjoint open neighborhoods。",
      "full finite audit遍歷全部unordered pairs。",
      "sample與finite-space corollary不外推。"
    ],
    "common_misconception_zh": "找到幾對可分點便稱Hausdorff，或把T1等同Hausdorff。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪positive general certificate限complete finite topology。",
    "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Hausdorff空間中的compact closedness。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-hausdorff-space.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-11-11 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"
  }
}