{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-continuous-image-of-compact",
  "semantic_family_id": "sf-continuous-image-of-compact",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "緊緻集的連續像"
  },
  "summary": {
    "zh-Hant": "若K在X中compact且f:X→Y連續，則image f(K)在Y中compact。"
  },
  "statement": {
    "zh-Hant": "對f(K)的ambient-open cover取逆像，continuity使逆像open且覆蓋K；K的finite subcover再推回覆蓋f(K)。不需要injective或surjective。compact image不反推domain compact；compact preimage或image closed也各需額外假設（後者常需Hausdorff）。"
  },
  "formula": {
    "tex": "C(f(K))",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mrow><mi>f</mi><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "function_call",
          "callee": {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          "arguments": [
            {
              "type": "symbol",
              "name": "K",
              "glyph": "K"
            }
          ]
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7b852887bdf08693377ccba9868e8f3ae2093d3adc09d44e46ff5d2015b7ab4c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "正向families使用完整有限來源與目標拓撲、total map及完整open covers。"
    },
    {
      "id": "a2",
      "zh-Hant": "K採subspace compactness；cover sets屬target topology。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "compactness_predicate",
      "role_zh": "緊緻性述詞",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "continuous_total_map",
      "role_zh": "連續全函數",
      "scope": "representative_formula"
    },
    {
      "token": "K",
      "role": "compact_subset",
      "role_zh": "緊緻子集",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-compact-subset",
      "title_zh": "緊緻子集",
      "reason_zh": "source compactness。"
    },
    {
      "id": "mko-topological-continuity",
      "title_zh": "拓撲連續性",
      "reason_zh": "open-preimage property。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數與映射",
      "reason_zh": "total image/preimage。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "image cover拉回為K的open cover。",
      "finite subcover indices原樣推回target cover。",
      "定理只有forward direction。"
    ],
    "common_misconception_zh": "由image compact反推domain compact，或無Hausdorff假設就稱image closed。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "finite positive certificates不外推所有infinite topology；反例另以standard open interval constant map表達。",
    "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Hausdorff closedness與quotient topology。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-continuous-image-of-compact.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-11-10 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"
  }
}