{"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-neighborhood"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-neighborhood-01","title":{"zh-Hant":"鄰域：在四點拓撲中搜尋開集見證並列出全部鄰域","en":"Neighborhood: searching for open witnesses and listing all neighborhoods in a four-point topology"},"summary":{"zh-Hant":"設 X={1,2,3,4}，τ={∅,{1},{1,2},{3,4},{1,3,4},X}。判定 N1={1,2,3} 與 N2={2,3,4} 是否為點 2 的鄰域；證明在有限拓撲中，點 x 的鄰域恰是包含 O_x（含 x 的全部開集之交）的子集，據此列出點 1、2、3、4 的全部鄰域；證明鄰域向上封閉；並說明同一集合在離散拓撲下的判定可能不同。","en":"Let X={1,2,3,4} and τ={∅,{1},{1,2},{3,4},{1,3,4},X}. Decide whether N1={1,2,3} and N2={2,3,4} are neighborhoods of the point 2; prove that in a finite topology the neighborhoods of x are exactly the subsets containing O_x (the intersection of all open sets containing x), and use this to list all neighborhoods of the points 1, 2, 3 and 4; prove that neighborhoods are closed upwards; and show that the verdict for the same set can change under the discrete topology."},"mko_ids":["mko-neighborhood"]}],"title":"Concept derivation guides"}