{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-path-connected-space",
  "semantic_family_id": "sf-path-connected-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "道路連通空間"
  },
  "summary": {
    "zh-Hant": "空間中任意兩點若可由單位區間的連續映射連接，則空間道路連通。"
  },
  "statement": {
    "zh-Hant": "path是continuous gamma:[0,1]→X且gamma(0)=x、gamma(1)=y。exact families以實interval與finite-dimensional box的linear path、兩條path的piecewise concatenation核endpoint、range與接點；disconnected gap與兩點discrete space用connected-image責任排除。只給有限sample points、只對某一對端點有path或忽略continuity都不證path connected。path connected推出connected，但一般converse不成立。"
  },
  "formula": {
    "tex": "H(X,x,y)=g",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>H</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "H",
          "glyph": "H"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "X",
            "glyph": "X"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          },
          {
            "type": "symbol",
            "name": "y",
            "glyph": "y"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "g",
        "glyph": "g"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "fc1135b5c82807d95a9efe691a8e00d0c31e3ea7959f5ad85c6f92d25525b343"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "domain固定為帶通常拓撲的閉單位區間。"
    },
    {
      "id": "a2",
      "zh-Hant": "每條candidate path須同時核連續性、endpoint與全range containment。"
    }
  ],
  "symbols": [
    {
      "token": "H",
      "role": "path_connectedness_operator",
      "role_zh": "道路連通性算子",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "topological_space",
      "role_zh": "拓撲空間",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "path_start",
      "role_zh": "路徑起點",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "path_end",
      "role_zh": "路徑終點",
      "scope": "representative_formula"
    },
    {
      "token": "g",
      "role": "continuous_path",
      "role_zh": "連續路徑",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-continuity",
      "title_zh": "拓撲連續性",
      "reason_zh": "path continuity。"
    },
    {
      "id": "mko-connected-topological-space",
      "title_zh": "連通拓撲空間",
      "reason_zh": "path connected implies connected與反向邊界。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "unit interval。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "0≤t≤1與interval containment。"
    },
    {
      "id": "mko-finite-product-topology",
      "title_zh": "有限乘積拓撲",
      "reason_zh": "box path的product codomain。"
    },
    {
      "id": "mko-intermediate-value-theorem",
      "title_zh": "介值定理",
      "reason_zh": "gap與discrete boundary的connected-image證書。"
    },
    {
      "id": "mko-composition",
      "title_zh": "函數合成",
      "reason_zh": "path concatenation與continuous image。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "算術運算",
      "reason_zh": "linear path與piecewise參數化。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "endpoint、continuity與全range containment缺一不可。",
      "convex interval／box給明示linear path，concatenation另驗接點。",
      "一對點或有限samples不證全稱path connectedness。"
    ],
    "common_misconception_zh": "把點列、圖上的walk或某一對點可連接當成拓撲path。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "linear exact witnesses只證指定convex spaces；一般connected不推出path connected。",
    "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local path connectedness與fundamental group入口。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-path-connected-space.md 整理；2026-10-15 脈衝 R06（使用者手動接續 2026-11-17 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}