{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-topological-continuity",
  "semantic_family_id": "sf-topological-continuity",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "拓撲連續映射"
  },
  "summary": {
    "zh-Hant": "拓撲映射連續，表示目標空間每個開集的逆像在來源空間仍是開集。"
  },
  "statement": {
    "zh-Hant": "給定拓撲空間 (X,τX)、(Y,τY) 與總映射 f:X→Y，f 連續當且僅當對每個 V∈τY，逆像 f⁻¹[V]∈τX。這是逆像條件，不要求開集的像為開。"
  },
  "formula": {
    "tex": "f:X\\to Y",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow></math>",
    "semantic_ast": {
      "type": "mapping",
      "function": {
        "type": "symbol",
        "name": "f",
        "glyph": "f"
      },
      "domain": {
        "type": "symbol",
        "name": "X",
        "glyph": "X"
      },
      "codomain": {
        "type": "symbol",
        "name": "Y",
        "glyph": "Y"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "28c28a16d0b842cb3d3c986cead38d14c1fbddc06efeb3c29e6e3479896c8a21"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "來源、目標底集合與兩個拓撲均固定。"
    },
    {
      "id": "a2",
      "zh-Hant": "f 是完整總表；逆像按目標子集合收集所有映入其中的來源元素。"
    }
  ],
  "symbols": [
    {
      "token": "f",
      "role": "total_map",
      "role_zh": "拓撲空間間的總映射",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "source_space",
      "role_zh": "來源空間",
      "scope": "representative_formula"
    },
    {
      "token": "Y",
      "role": "target_space",
      "role_zh": "目標空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "來源與目標各有開集族。"
    },
    {
      "id": "mko-open-closed-set",
      "title_zh": "開集與閉集",
      "reason_zh": "連續性核對目標開集及來源逆像。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "逆像由完整總映射定義。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "有限拓撲中可以逐一列出目標所有開集並重算逆像，因此完整判定給定表是否連續。",
      "連續映射的合成仍連續；有限模型可重算兩段映射與每個開集逆像，不能只抽樣單點。",
      "開映射要求開集的像為開，與連續性不同。映射可能連續但不是開映射，也可能是開映射但不連續。"
    ],
    "common_misconception_zh": "把開集的像為開當成連續定義，或只抽查一個開集。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "完整有限拓撲可精確判定；不宣稱既有實函數連續條目的所有語義已自動等價。",
    "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 Continuous 的 open-preimage 定義，另建立實函數、子空間與孤立點語義橋。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-topological-continuity.md 整理；2026-10-13 脈衝 R03（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"
  }
}