{"schema_version":"ocme-site-results-v1","locale":"en","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-topological-continuity"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-topological-continuity-01","title":{"zh-Hant":"拓撲連續映射：在有限空間中用逆像判定連續，並與開映射區分","en":"Continuous maps between topological spaces: deciding continuity by preimages in finite spaces and separating it from openness"},"summary":{"zh-Hant":"設 X={1,2,3}，τX={∅,{1},{1,2},X}；Y={a,b}，τY={∅,{a},Y}。對三個以完整表格給出的映射 f（1↦a、2↦a、3↦b）、g（1↦a、2↦b、3↦a）、c（全部 ↦b），用逆像條件判定是否連續，並判定是否為開映射；證明連續映射的合成仍連續並以 q:Y→Z 驗證；最後數出 X→Y 的全部連續映射。","en":"Let X={1,2,3} with τX={∅,{1},{1,2},X}, and Y={a,b} with τY={∅,{a},Y}. For three maps given by complete tables, f (1↦a, 2↦a, 3↦b), g (1↦a, 2↦b, 3↦a) and c (everything ↦b), decide continuity by the preimage condition and decide whether each is an open map; prove that a composite of continuous maps is continuous and check it with q:Y→Z; finally count all continuous maps X→Y."},"mko_ids":["mko-topological-continuity"]}],"title":"Concept derivation guides"}