{"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-open-closed-set"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-open-closed-set-01","title":{"zh-Hant":"開集與閉集：在四點拓撲中完整分類十六個子集","en":"Open and closed sets: classifying all sixteen subsets of a four-point topology"},"summary":{"zh-Hant":"設 X={1,2,3,4}，τ={∅,{1},{1,2},{3,4},{1,3,4},X}。先驗證 τ 是 X 上的拓撲，再列出全部閉集，並把 X 的十六個子集分成「既開又閉」「只開」「只閉」「既不開也不閉」四類；最後證明在任何拓撲空間中 ∅ 與 X 都既開又閉，以及兩個閉集的聯集仍是閉集。","en":"Let X={1,2,3,4} and τ={∅,{1},{1,2},{3,4},{1,3,4},X}. First verify that τ is a topology on X, then list all closed sets and sort the sixteen subsets of X into four classes: clopen, open only, closed only, and neither; finally prove that in every topological space ∅ and X are clopen, and that the union of two closed sets is closed."},"mko_ids":["mko-open-closed-set"]}],"title":"概念推導導讀"}