{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-neighborhood",
  "semantic_family_id": "sf-neighborhood",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "鄰域"
  },
  "summary": {
    "zh-Hant": "集合 N 是點 x 的鄰域，當且僅當 N 包含某個含 x 的開集；N 本身不必開。"
  },
  "statement": {
    "zh-Hant": "在拓撲空間 (X,τ) 中，N⊆X 是 x∈X 的鄰域，當且僅當存在 U∈τ，使 x∈U⊆N。所有含 x 的開集構成 x 的開鄰域基礎，但一般鄰域本身不必開。"
  },
  "formula": {
    "tex": "x\\in N",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>x</mi><mo>∈</mo><mi>N</mi></mrow></math>",
    "semantic_ast": {
      "type": "membership",
      "element": {
        "type": "symbol",
        "name": "x",
        "glyph": "x"
      },
      "set": {
        "type": "symbol",
        "name": "N",
        "glyph": "N"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "34fdda6b73ebb8d2d16f0bc7a0d43be6155c0549b6202b69c3d00f6c0cff5b1f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "X、τ、x 與候選 N 已明示，且 x∈X、N⊆X。"
    },
    {
      "id": "a2",
      "zh-Hant": "鄰域採包含開集的集合語義；不預設距離、半徑或單一最小鄰域。"
    }
  ],
  "symbols": [
    {
      "token": "x",
      "role": "point",
      "role_zh": "指定點",
      "scope": "representative_formula"
    },
    {
      "token": "N",
      "role": "neighborhood_candidate",
      "role_zh": "候選鄰域",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "鄰域由開集族導出。"
    },
    {
      "id": "mko-open-closed-set",
      "title_zh": "開集與閉集",
      "reason_zh": "見證 U 必須是開集。"
    },
    {
      "id": "mko-set-membership",
      "title_zh": "集合隸屬",
      "reason_zh": "指定點需屬於開集見證與候選鄰域。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "核對 N 是否為鄰域時，應搜尋所有含 x 且包含於 N 的開集見證。N 自己不是開集也可能是鄰域。",
      "鄰域具有向上封閉性：若 N 是 x 的鄰域且 N⊆M⊆X，則 M 也是 x 的鄰域。",
      "有限拓撲可完整列出某點的全部鄰域；這個列表依賴拓撲與點，不能只看底集合。"
    ],
    "common_misconception_zh": "把鄰域等同於開集或開球，或只檢查 x∈N 就宣稱 N 是鄰域。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限冪集枚舉可完整核對指定模型；不外推度量球或一般濾子定理。",
    "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 nhds、nhds_basis_opens 與 neighborhood filter 的向上封閉語義。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-neighborhood.md 整理；2026-10-13 脈衝 R02（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"
  }
}