{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"paths","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":{},"path":{"id":"path-pulse-foundations-20261019","title":{"zh-Hant":"擴展數學地圖第二批：拓撲語言、計算正確性、漸近界與證明項","en":"Second Extended Atlas Slice: Topological Language, Computational Correctness, Asymptotic Bounds, and Proof Terms"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proof","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-lean-formal-statement","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-open-closed-set","position":2,"requirement":"optional","rationale":{"zh-Hant":"在固定拓撲空間中，開集屬於拓撲；閉集的補集為開集，同一集合可以同時開且閉。","en":"In a fixed topological space, open sets belong to the topology and closed sets have open complements; one set can be both open and closed."}},{"id":"mko-neighborhood","position":3,"requirement":"optional","rationale":{"zh-Hant":"集合 N 是點 x 的鄰域，當且僅當 N 包含某個含 x 的開集；N 本身不必開。","en":"A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open."}},{"id":"mko-topological-continuity","position":4,"requirement":"optional","rationale":{"zh-Hant":"拓撲映射連續，表示目標空間每個開集的逆像在來源空間仍是開集。","en":"A map is topologically continuous when every open set in its target has an open preimage in its source."}},{"id":"mko-computational-model","position":5,"requirement":"optional","rationale":{"zh-Hant":"有限狀態計算模型以狀態集、初始狀態集、步驟關係與停止狀態集描述可能執行。","en":"A finite-state computational model describes possible executions through its state set, initial-state set, step relation and halt-state set."}},{"id":"mko-algorithm-correctness","position":6,"requirement":"optional","rationale":{"zh-Hant":"算法正確性分開部分正確性與終止性；總正確性需要兩者同時成立。","en":"Algorithm correctness separates partial correctness from termination; total correctness requires both."}},{"id":"mko-asymptotic-resource-bound","position":7,"requirement":"optional","rationale":{"zh-Hant":"漸近上界、下界與緊界必須相對輸入大小、成本模型及全尾段量化定義。","en":"Asymptotic upper, lower and tight bounds require an input-size convention, cost model and quantification over a full tail."}},{"id":"mko-lean-proof-term","position":8,"requirement":"optional","rationale":{"zh-Hant":"證明項是在固定形式環境與 context 中具有命題 P 型別，並可由 kernel 檢查的項 t。","en":"A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel."}}]},"title":"Second Extended Atlas Slice: Topological Language, Computational Correctness, Asymptotic Bounds, and Proof Terms"}