See topological space in All languages combined, or Wiktionary
{ "forms": [ { "form": "topological spaces", "tags": [ "plural" ] } ], "head_templates": [ { "args": {}, "expansion": "topological space (plural topological spaces)", "name": "en-noun" } ], "lang": "English", "lang_code": "en", "pos": "noun", "senses": [ { "categories": [ { "kind": "topical", "langcode": "en", "name": "Topology", "orig": "en:Topology", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "47 53", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "46 54", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "45 55", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" } ], "examples": [ { "ref": "1964, H. J. Kowalsky, translated by Jay E. Strum, Topological Spaces, Academic Press, page 165:", "text": "In order to obtain \"intuitive insight\" into special classes of topological spaces we can proceed in several ways, only a few of which will be pursued in this chapter. For instance, we can seek to describe important topological spaces by means of enough of their properties to completely characterize them, up to homeomorphism.", "type": "quote" }, { "text": "2011, Jonathan A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Springer, Lecture Notes in Mathematics 2032, page xi,\nMost of the spaces studied in Algebraic Topology, such as CW-complexes or manifolds, are Hausdorff. In contrast, finite topological spaces are rarely Hausdorff. A topological space with finitely many points, each of which is closed, must be discrete." }, { "ref": "2013, D. J. H. Garling, A Course in Mathematical Analysis: Volume 2, Cambridge University Press, page 431:", "text": "If (X,#x5C;tau) is a topological space, then a cover #x5C;mathcalB is open if each B#x5C;in#x5C;mathcalB is an open set. A topological space (X,#x5C;tau) is compact if every open cover of X has a finite subcover.", "type": "quote" } ], "glosses": [ "An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);" ], "id": "en-topological_space-en-noun-Q542B8kx", "links": [ [ "topology", "topology" ], [ "ordered pair", "ordered pair" ], [ "set", "set" ], [ "open set", "open set" ], [ "closed set", "closed set" ] ], "raw_glosses": [ "(topology, formally) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);" ], "tags": [ "formal" ], "topics": [ "mathematics", "sciences", "topology" ] }, { "categories": [ { "kind": "topical", "langcode": "en", 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"other", "name": "Terms with Japanese translations", "parents": [], "source": "w+disamb" }, { "_dis": "40 60", "kind": "other", "name": "Terms with Mandarin translations", "parents": [], "source": "w+disamb" }, { "_dis": "41 59", "kind": "other", "name": "Terms with Portuguese translations", "parents": [], "source": "w+disamb" }, { "_dis": "39 61", "kind": "other", "name": "Terms with Romanian translations", "parents": [], "source": "w+disamb" }, { "_dis": "39 61", "kind": "other", "name": "Terms with Russian translations", "parents": [], "source": "w+disamb" }, { "_dis": "39 61", "kind": "other", "name": "Terms with Serbo-Croatian translations", "parents": [], "source": "w+disamb" }, { "_dis": "40 60", "kind": "other", "name": "Terms with Spanish translations", "parents": [], "source": "w+disamb" }, { "_dis": "39 61", "kind": "other", "name": "Terms with Swedish translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 64", "kind": "other", "langcode": "en", "name": "Topological spaces", "orig": "en:Topological spaces", "parents": [], "source": "w+disamb" } ], "glosses": [ "An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);\n(loosely) the set X.", "the set X." ], "hypernyms": [ { "_dis1": "41 59", "word": "mathematical space" }, { "_dis1": "41 59", "word": "space" } ], "hyponyms": [ { "_dis1": "36 64", "sense": "uniform space", "word": "metric space" }, { "_dis1": "18 82", "word": "normal space" }, { "_dis1": "18 82", "word": "pointed space" }, { "_dis1": "18 82", "word": "regular space" }, { "_dis1": "18 82", "word": "suspension" }, { "_dis1": "35 65", "sense": "sober space", "word": "Hausdorff space" } ], "id": "en-topological_space-en-noun-akpPmCYj", "links": [ [ "topology", "topology" ], [ "ordered pair", "ordered pair" ], [ "set", "set" ], [ "open set", "open set" ], [ "closed set", "closed set" ] ], "meronyms": [ { "_dis1": "18 82", "word": "topology" }, { "_dis1": "18 82", "word": "open set" }, { "_dis1": "18 82", "word": "underlying set" } ], "raw_glosses": [ "(topology, formally) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);" ], "tags": [ "broadly", "formal" ], "topics": [ "mathematics", "sciences", "topology" ], "translations": [ { "_dis1": "40 60", "code": "cmn", "lang": "Chinese Mandarin", "roman": "tuòpū kōngjiān", "sense": "set that has a topology", "word": "拓扑空间" }, { "_dis1": "40 60", "code": "da", "lang": "Danish", "sense": "set that has a topology", "tags": [ "neuter" ], "word": "topologisk rum" }, { "_dis1": "40 60", "code": "fi", "lang": "Finnish", "sense": "set that has a topology", "word": "topologinen avaruus" }, { "_dis1": "40 60", "code": "gl", "lang": "Galician", "sense": "set that has a topology", "tags": [ "masculine" ], "word": "espazo topolóxico" }, { "_dis1": "40 60", "code": "el", "lang": "Greek", "roman": "topologikós chóros", "sense": "set that has a topology", "tags": [ "masculine" ], "word": "τοπολογικός χώρος" }, { "_dis1": "40 60", "code": "it", "lang": "Italian", "sense": "set that has a topology", "tags": [ "masculine" ], "word": "spazio topologico" }, { "_dis1": "40 60", "code": "sv", "lang": "Swedish", "sense": "set that has a topology", "word": "topologiskt rum" } ] } ], "translations": [ { "_dis1": "50 50", "code": "af", "lang": "Afrikaans", "sense": "ordered pair of a set and its topology", "word": "topologiese ruimte" }, { "_dis1": "50 50", "code": "cmn", "lang": "Chinese Mandarin", "roman": "tuòpū kōngjiān", "sense": "ordered pair of a set and its topology", "word": "拓扑空间" }, { "_dis1": "50 50", "code": "da", "lang": "Danish", "sense": "ordered pair of a set and its topology", "tags": [ "neuter" ], "word": "topologisk rum" }, { "_dis1": "50 50", "code": "fi", "lang": "Finnish", "sense": "ordered pair of a set and its topology", "word": "topologinen avaruus" }, { "_dis1": "50 50", "code": "fr", "lang": "French", "sense": "ordered pair of a set and its topology", "tags": [ "feminine" ], "word": "espace topologique" }, { "_dis1": "50 50", "code": "gl", "lang": "Galician", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "espazo tipolóxico" }, { "_dis1": "50 50", "code": "de", "lang": "German", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "topologischer Raum" }, { "_dis1": "50 50", "code": "el", "lang": "Greek", "roman": "topologikós chóros", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "τοπολογικός χώρος" }, { "_dis1": "50 50", "code": "he", "lang": "Hebrew", "roman": "mercháv topológi", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "מרחב טופולוגי" }, { "_dis1": "50 50", "code": "is", "lang": "Icelandic", "sense": "ordered pair of a set and its topology", "tags": [ "neuter" ], "word": "grannrúm" }, { "_dis1": "50 50", "code": "it", "lang": "Italian", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "spazio topologico" }, { "_dis1": "50 50", "alt": "いそう くうかん", "code": "ja", "lang": "Japanese", "roman": "isō kūkan", "sense": "ordered pair of a set and its topology", "word": "位相空間" }, { "_dis1": "50 50", "code": "pt", "lang": "Portuguese", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "espaço topológico" }, { "_dis1": "50 50", "code": "ro", "lang": "Romanian", "sense": "ordered pair of a set and its topology", "tags": [ "neuter" ], "word": "spațiu topologic" }, { "_dis1": "50 50", "code": "ru", "lang": "Russian", "roman": "topologíčeskoje prostránstvo", "sense": "ordered pair of a set and its topology", "tags": [ "neuter" ], "word": "топологи́ческое простра́нство" }, { "_dis1": "50 50", "code": "es", "lang": "Spanish", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "espacio topológico" }, { "_dis1": "50 50", "code": "sv", "lang": "Swedish", "sense": "ordered pair of a set and its topology", "word": "topologiskt rum" } ], "wikipedia": [ "topological space" ], "word": "topological space" }
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J. Kowalsky, translated by Jay E. Strum, Topological Spaces, Academic Press, page 165:", "text": "In order to obtain \"intuitive insight\" into special classes of topological spaces we can proceed in several ways, only a few of which will be pursued in this chapter. For instance, we can seek to describe important topological spaces by means of enough of their properties to completely characterize them, up to homeomorphism.", "type": "quote" }, { "text": "2011, Jonathan A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Springer, Lecture Notes in Mathematics 2032, page xi,\nMost of the spaces studied in Algebraic Topology, such as CW-complexes or manifolds, are Hausdorff. In contrast, finite topological spaces are rarely Hausdorff. A topological space with finitely many points, each of which is closed, must be discrete." }, { "ref": "2013, D. J. H. 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A topological space (X,#x5C;tau) is compact if every open cover of X has a finite subcover.", "type": "quote" } ], "glosses": [ "An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);" ], "links": [ [ "topology", "topology" ], [ "ordered pair", "ordered pair" ], [ "set", "set" ], [ "open set", "open set" ], [ "closed set", "closed set" ] ], "raw_glosses": [ "(topology, formally) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);" ], "tags": [ "formal" ], "topics": [ "mathematics", "sciences", "topology" ] }, { "categories": [ "English terms with quotations", "en:Topology" ], "glosses": [ "An ordered pair (X, 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kōngjiān", "sense": "ordered pair of a set and its topology", "word": "拓扑空间" }, { "code": "da", "lang": "Danish", "sense": "ordered pair of a set and its topology", "tags": [ "neuter" ], "word": "topologisk rum" }, { "code": "fi", "lang": "Finnish", "sense": "ordered pair of a set and its topology", "word": "topologinen avaruus" }, { "code": "fr", "lang": "French", "sense": "ordered pair of a set and its topology", "tags": [ "feminine" ], "word": "espace topologique" }, { "code": "gl", "lang": "Galician", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "espazo tipolóxico" }, { "code": "de", "lang": "German", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "topologischer Raum" }, { "code": "el", "lang": "Greek", "roman": "topologikós chóros", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "τοπολογικός χώρος" }, { "code": "he", "lang": "Hebrew", "roman": "mercháv topológi", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "מרחב טופולוגי" }, { "code": "is", "lang": "Icelandic", "sense": "ordered pair of a set and its topology", "tags": [ "neuter" ], "word": "grannrúm" }, { "code": "it", "lang": "Italian", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "spazio topologico" }, { "alt": "いそう くうかん", "code": "ja", "lang": "Japanese", "roman": "isō kūkan", "sense": "ordered pair of a set and its topology", "word": "位相空間" }, { "code": "pt", "lang": "Portuguese", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "espaço topológico" }, { "code": "ro", "lang": "Romanian", "sense": "ordered pair of a set and its topology", "tags": [ "neuter" ], "word": "spațiu topologic" }, { "code": "ru", "lang": "Russian", "roman": "topologíčeskoje prostránstvo", "sense": "ordered pair of a set and its topology", "tags": [ "neuter" ], "word": "топологи́ческое простра́нство" }, { "code": "es", "lang": "Spanish", "sense": "ordered pair of a set and its topology", "tags": [ "masculine" ], "word": "espacio topológico" }, { "code": "sv", "lang": "Swedish", "sense": "ordered pair of a set and its topology", "word": "topologiskt rum" }, { "code": "cmn", "lang": "Chinese Mandarin", "roman": "tuòpū kōngjiān", "sense": "set that has a topology", "word": "拓扑空间" }, { "code": "da", "lang": "Danish", "sense": "set that has a topology", "tags": [ "neuter" ], "word": "topologisk rum" }, { "code": "fi", "lang": "Finnish", "sense": "set that has a topology", "word": "topologinen avaruus" }, { "code": "gl", "lang": "Galician", "sense": "set that has a topology", "tags": [ "masculine" ], "word": "espazo topolóxico" }, { "code": "el", "lang": "Greek", "roman": "topologikós chóros", "sense": "set that has a topology", "tags": [ "masculine" ], "word": "τοπολογικός χώρος" }, { "code": "it", "lang": "Italian", "sense": "set that has a topology", "tags": [ "masculine" ], "word": "spazio topologico" }, { "code": "sv", "lang": "Swedish", "sense": "set that has a topology", "word": "topologiskt rum" } ], "wikipedia": [ "topological space" ], "word": "topological space" }
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This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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