{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:d89858ae-0ae4-44a6-b482-787218d5e179", "labels": { "zh-CN": "边界", "en": "boundary" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "8442909028fc04ec8e49ea334a21e9e990b3fdce85e1ca8b6b66d3e16053fe43", "occurrences": [ { "concept_id": "urn:uuid:d89858ae-0ae4-44a6-b482-787218d5e179", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-10/02-general-topological-spaces", "anchor": "first-ccd810e4a97cf523defc-zh", "chapter": "10", "section": "10.2", "offset": 13735, "range": [ 20756, 20987 ], "source_revision": "a1c67f92b8f8c8fbb6719a655bbf4fea791f002ad179f1ac69f92336bc7fccfb", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "边界(boundary)", "context": { "range": [ 20132, 20988 ], "quote": "任意集合 $A$ 的 闭包(closure)，记作 $\\overline{A}$，是集合 $A$ 与其所有聚点的并集。$A$ 的 内部(interior) 是所有开集 $U \\subseteq A ,$ 的并集，记作 $A ^ { o }$。这两个集合的差 $b ( A ) = { \\overline { { A } } } - A ^ { o }$ 称为 $A$ 的 边界(boundary)。", "quote_sha256": "a38d553d30b7e866297871639696e0a554b843812d3b0caf37b6850125350a3e", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-10/02-general-topological-spaces", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-e69ce63417e83b75d39b7303" }, { "concept_id": "urn:uuid:d89858ae-0ae4-44a6-b482-787218d5e179", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-17/04-homology-and-cohomology", "anchor": "exercise-17-4-1", "chapter": "17", "section": "17.4", "offset": 7404, "range": [ 11701, 11897 ], "source_revision": "962da89618680522177106de9b1899c86eb3dedc9484f12c33659d432388a117", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "边界(boundary)", "context": { "range": [ 11688, 12258 ], "quote": "一个$p$-链$C$若没有边界(boundary)，即$\\partial C = 0 .$，则称其为循环(cycle)。若存在一个$(p+1)$-链$C^{\\prime}$使得$C = \\partial C ^ { \\prime }$，则称其为边界。显然，每个边界都是$p$-循环，因为$\\partial C = \\partial ^ { 2 } C ^ { \\prime } = 0$，但反之未必成立。", "quote_sha256": "87ab1013623f7b1d866eec5fd7e2263c4feb3674f6500fd026699ca6e6470ef7", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-17/04-homology-and-cohomology", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-5c981ac1a91cb8be7b4adc1f" }, { "concept_id": "urn:uuid:d89858ae-0ae4-44a6-b482-787218d5e179", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-17/04-homology-and-cohomology", "anchor": "first-ccd810e4a97cf523defc-en", "chapter": "17", "section": "17.4", "offset": 9410, "range": [ 13234, 13461 ], "source_revision": "bcdaffd9c93fa5d4dbb16ac427fef782512ff58c5d5472ea282c7769a2173a3d", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "boundary", "context": { "range": [ 12964, 13730 ], "quote": "A $p$-chain $C$ is said to be a cycle if it has no boundary, $\\partial C = 0 .$ . 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