See topological space on Wiktionary
Download JSON data for topological space meaning in All languages combined (8.6kB)
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"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.",
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"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.",
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"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.",
"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);"
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"word": "pointed space"
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"word": "regular space"
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"word": "suspension"
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"sense": "sober space",
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"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "tuòpū kōngjiān",
"sense": "set that has a topology",
"word": "拓扑空间"
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"word": "espazo topolóxico"
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"code": "el",
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"sense": "set that has a topology",
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"masculine"
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"word": "τοπολογικός χώρος"
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"sense": "ordered pair of a set and its topology",
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{
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"code": "el",
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"sense": "ordered pair of a set and its topology",
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"word": "τοπολογικός χώρος"
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"roman": "mercháv topológi",
"sense": "ordered pair of a set and its topology",
"tags": [
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],
"word": "מרחב טופולוגי"
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{
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"code": "is",
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"sense": "ordered pair of a set and its topology",
"tags": [
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"word": "grannrúm"
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"_dis1": "50 50",
"code": "it",
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"sense": "ordered pair of a set and its topology",
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"word": "spazio topologico"
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"_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",
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"sense": "ordered pair of a set and its topology",
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"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": "quotation"
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{
"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": "quotation"
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"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.",
"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);"
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"(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);\n"
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],
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"code": "af",
"lang": "Afrikaans",
"sense": "ordered pair of a set and its topology",
"word": "topologiese ruimte"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "tuòpū 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": [
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],
"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": [
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],
"word": "מרחב טופולוגי"
},
{
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"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",
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"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": [
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],
"word": "topologisk rum"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "set that has a topology",
"word": "topologinen avaruus"
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{
"code": "gl",
"lang": "Galician",
"sense": "set that has a topology",
"tags": [
"masculine"
],
"word": "espazo topolóxico"
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{
"code": "el",
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"roman": "topologikós chóros",
"sense": "set that has a topology",
"tags": [
"masculine"
],
"word": "τοπολογικός χώρος"
},
{
"code": "it",
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"word": "spazio topologico"
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"sense": "set that has a topology",
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],
"wikipedia": [
"topological space"
],
"word": "topological space"
}
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-05-01 from the enwiktionary dump dated 2024-04-21 using wiktextract (f4fd8c9 and c9440ce). 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.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.