{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-topological-space",
  "semantic_family_id": "sf-topological-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "拓撲空間"
  },
  "summary": {
    "zh-Hant": "拓撲空間由底集合與一族開集組成；開集族包含空集與全空間，並對任意聯集與有限交封閉。"
  },
  "statement": {
    "zh-Hant": "給定集合 X，X 上的拓撲是子集合族 τ⊆P(X)，滿足 ∅∈τ、X∈τ、τ 中任意子族的聯集仍在 τ，且 τ 中任意有限子族的交集仍在 τ；(X,τ) 稱為拓撲空間。"
  },
  "formula": {
    "tex": "T\\subseteq P",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>T</mi><mo>⊆</mo><mi>P</mi></mrow></math>",
    "semantic_ast": {
      "type": "set_subset",
      "subset": {
        "type": "symbol",
        "name": "T",
        "glyph": "T"
      },
      "superset": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7c47455a830ebe8c31501fda72e922e1ada2e76000ca2fdea824e79352ac9cff"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "底集合 X 已固定；τ 的每個成員都是 X 的子集合。"
    },
    {
      "id": "a2",
      "zh-Hant": "任意聯集包含空族情形，有限交包含空族情形；空族交集按 X 解讀。"
    }
  ],
  "symbols": [
    {
      "token": "T",
      "role": "topology_family",
      "role_zh": "代表開集族 τ",
      "scope": "representative_formula"
    },
    {
      "token": "P",
      "role": "power_set",
      "role_zh": "代表底集合 X 的冪集 P(X)",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "底集合與開集族都以集合語義表達。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "拓撲公理使用聯集、有限交集、空集與全空間。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "拓撲指定哪些子集合被稱為開集；它不先指定距離。離散拓撲把所有子集合都列為開集，不可分拓撲只包含空集與全空間。",
      "有限底集合上可以完整枚舉候選子集合族並檢查三類封閉條件；只抽查幾組聯集或交集不足以證明候選族是拓撲。",
      "同一底集合可帶不同拓撲。比較粗細時，τ₁⊆τ₂ 表示 τ₂ 較細；這是開集族的包含關係，不是底集合大小。"
    ],
    "common_misconception_zh": "把任何子集合族都當成拓撲，或把只對兩兩聯集封閉誤當任意聯集封閉。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限底集合可由完整冪集枚舉精確核對；這不取代一般任意聯集公理的形式證明。",
    "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 Mathlib TopologicalSpace 的 IsOpen、univ、二元交與 sUnion 公理，並建立有限枚舉語義橋。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-topological-space.md 整理；2026-10-13 脈衝 R01（使用者手動接續 2026-10-18 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"
  }
}