{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-metric-space"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-metric-space-01","title":{"zh-Hant":"度量空間：逐格與逐三元組核對有限距離表","en":"Metric space: auditing a finite distance table cell by cell and triple by triple"},"summary":{"zh-Hant":"給三張完整整數距離表。（甲）X={0,1,2,3} 上的表 d（四點環上的步數距離）：逐格核非負、對角線為零、非對角線不為零、對稱，並以涵蓋全部 4³=64 個三元組的分情形論證核三角不等式，得出 (X,d) 是度量空間。（乙）Y={0,1,2} 上的表 e 通過前四項卻在 (0,1,2) 違反三角不等式。（丙）Z={0,1,2} 上的表 f 非負、對稱、對角線為零，卻有非對角零值，違反分離性。","en":"Three complete integer distance tables are given. (A) On X={0,1,2,3}, table d (step distance on a 4-cycle): check nonnegativity, zero diagonal, nonzero off-diagonal entries and symmetry cell by cell, and the triangle inequality by a case argument covering all 4³=64 triples, concluding that (X,d) is a metric space. (B) Table e on Y={0,1,2} passes the first four checks but violates the triangle inequality at (0,1,2). (C) Table f on Z={0,1,2} is nonnegative, symmetric and zero on the diagonal, yet has an off-diagonal zero and violates separation."},"mko_ids":["mko-metric-space"]}],"title":"概念推導導讀"}