{
"schema_version": 1,
"kind": "ConceptView",
"producer": "concept-view-1",
"concept_id": "urn:uuid:defda17a-18a4-441f-bd21-63aeaa375b3f",
"labels": {
"zh-CN": "加细",
"en": "refinement"
},
"alternative_labels": [],
"scope_note": "",
"identity_status": "active",
"identity_revision": "0da2e5d187a9be08f1c9d903747c2d8bf83e8245275a41fbd4b333126f63fdd3",
"occurrences": [
{
"concept_id": "urn:uuid:defda17a-18a4-441f-bd21-63aeaa375b3f",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "en",
"route": "en/chapter-17/01-partitions-of-unity",
"anchor": "first-d9fa818df9de9f675c8b-en",
"chapter": "17",
"section": "17.1",
"offset": 465,
"range": [
3323,
3552
],
"source_revision": "7b9b5ac2d16a43bad55f3768f5df24ebc80a279e681dc057d54869725dff60c3",
"source_category": "original_transcription",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "refinement",
"context": {
"range": [
3189,
4560
],
"quote": "Given an open covering $\\{ U _ { i } | i \\in I \\}$ of a topological space $S ,$ an open covering $\\{ V _ { a } \\}$ of $S$ is called a refinement of $\\{ U _ { i } \\}$ if each $V _ { a } \\subseteq U _ { i }$ for some $i \\in I$ . The refinement is said to be locally finite if every point $p$ belongs to at most a finite number of $V _ { a } \\mathbf { \\bar { s } } .$ . The topological space $S$ is said to be paracompact if for every open covering $U _ { i } \\left( i \\in I \\right)$ there exists a locally finite refinement. As it may be shown that every locally compact Hausdorff secondcountable space is paracompact [12], we will from now on restrict attention to manifolds that are paracompact topological spaces.",
"quote_sha256": "2c7f46df4850b73f265175392d76dcec8f681ea31c69effe1305a56d2980ae26",
"environment": null,
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "zh/chapter-17/01-partitions-of-unity",
"lang": "zh-CN",
"source_category": "translation",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-cfafa189c9193f7ef2736aa9"
},
{
"concept_id": "urn:uuid:defda17a-18a4-441f-bd21-63aeaa375b3f",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "zh-CN",
"route": "zh/chapter-17/01-partitions-of-unity",
"anchor": "first-d9fa818df9de9f675c8b-zh",
"chapter": "17",
"section": "17.1",
"offset": 482,
"range": [
3811,
4044
],
"source_revision": "ac4949a5290e2d242874f181575e639623b84e0d6867c49e0b4130c31d8a7763",
"source_category": "translation",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "加细(refinement)",
"context": {
"range": [
3660,
4717
],
"quote": "给定拓扑空间 $S ,$ 的一个开覆盖 $\\{ U _ { i } | i \\in I \\}$，若对于每个 $i \\in I$ 都有 $V _ { a } \\subseteq U _ { i }$，则称 $S$ 的开覆盖 $\\{ V _ { a } \\}$ 为 $\\{ U _ { i } \\}$ 的 加细(refinement)。若每一点 $p$ 至多属于有限个 $V _ { a } \\mathbf { \\bar { s } } .$，则称该加细是 局部有限的(locally finite)。若对每个开覆盖 $U _ { i } \\left( i \\in I \\right)$ 都存在局部有限的加细，则称拓扑空间 $S$ 是 仿紧的(paracompact)。由于可以证明每个局部紧 Hausdorff 第二可数空间都是仿紧的 [12]，因此从现在起我们将注意力限制在作为仿紧拓扑空间的流形上。",
"quote_sha256": "c2ae93d4e9539c0b8d151b1d049fae57f1d797974198eea78268916d61b6122f",
"environment": null,
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "en/chapter-17/01-partitions-of-unity",
"lang": "en",
"source_category": "original_transcription",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-364d7ba002a3eae19306cde9"
}
],
"registered_positions": [
{
"route": "en/chapter-17/01-partitions-of-unity",
"lang": "en",
"section": "17.1",
"signal": "term-mark",
"label": "refinement",
"english": "refinement",
"offset": 465
},
{
"route": "zh/chapter-17/01-partitions-of-unity",
"lang": "zh-CN",
"section": "17.1",
"signal": "term-mark",
"label": "加细",
"english": "refinement",
"offset": 482
}
],
"statements": [],
"resolved_references": [],
"reading_order": [
"occ-cfafa189c9193f7ef2736aa9",
"occ-364d7ba002a3eae19306cde9"
],
"recorded_dependencies": [],
"reading_paths": [],
"coverage": {
"registered_positions": 2,
"located_occurrences": 2,
"whole_book_search_complete": false,
"registered_positions_truncated": false,
"independent_human_review_implied": false
},
"unresolved": [],
"base_revision": "e3331bc8ea42c62f9a8775f0eebe57f7d8ba46ce1d1a980f717764cc9587be56",
"optional_enrichments": []
}