{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-subspace-topology",
  "semantic_family_id": "sf-subspace-topology",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "子空間拓撲"
  },
  "summary": {
    "zh-Hant": "Y⊆X的相對拓撲由全部母空間open sets與Y的交集組成。"
  },
  "statement": {
    "zh-Hant": "tau_Y={U∩Y:U∈tau_X}，重複與空交集正規化。relative-open未必ambient-open且不要求U⊆Y；relative-closed以Y內補集open或F∩Y、F ambient closed判定。全部拓撲公理以Y作相對全集。"
  },
  "formula": {
    "tex": "T(Y)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>T</mi><mo>(</mo><mi>Y</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "T",
        "glyph": "T"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "Y",
          "glyph": "Y"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "eaab593b6bc30521b4f7e7be8d372b879d394581b8d623e5308367fa0fa01e54"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限X、完整母拓撲與明示subset Y。"
    },
    {
      "id": "a2",
      "zh-Hant": "候選relative sets必為Y的子集。"
    }
  ],
  "symbols": [
    {
      "token": "T",
      "role": "subspace_topology",
      "role_zh": "子空間拓撲",
      "scope": "representative_formula"
    },
    {
      "token": "Y",
      "role": "subspace_carrier",
      "role_zh": "子空間carrier",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "母拓撲。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "Y⊆X。"
    },
    {
      "id": "mko-topological-basis",
      "title_zh": "拓撲基底",
      "reason_zh": "basis intersection構造。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "每個relative open是O∩Y。",
      "complement在Y內取。",
      "relative與ambient openness不可混同。"
    ],
    "common_misconception_zh": "要求O⊆Y或把relative-open一律當X中的open。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "complete enumeration只證明宣告有限拓撲。",
    "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compact subsets與product topology。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-subspace-topology.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-11-08 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"
  }
}