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| Coarse structure ： ウィキペディア英語版 |
:''"Coarse space" redirects here. For the use of "coarse space" in numerical analysis, see coarse problem.''In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined.The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.==Definition==A coarse structure on a set ''X'' is a collection E of subsets of ''X'' × ''X'' (therefore falling under the more general categorization of binary relations on ''X'') called ''controlled sets'', and so that E possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:;1. Identity/diagonal: The diagonal Δ = is a member of E—the identity relation.;2. Closed under taking subsets: If ''E'' is a member of E and ''F'' is a subset of ''E'', then ''F'' is a member of E.;3. Closed under taking inverses: If ''E'' is a member of E then the inverse (or transpose) ''E'' −1 = is a member of E—the inverse relation.;4. Closed under taking unions: If ''E'' and ''F'' are members of E then the union of ''E'' and ''F'' is a member of E.;5. Closed under composition: If ''E'' and ''F'' are members of E then the product ''E'' o ''F'' = is a member of E—the composition of relations.A set ''X'' endowed with a coarse structure E is a ''coarse space''.The set ''E''() is defined as . We define the ''section'' of ''E'' by ''x'' to be the set ''E''(), also denoted ''E'' ''x''. The symbol ''E'y'' denotes the set ''E'' −1(). These are forms of projections.
:''"Coarse space" redirects here. For the use of "coarse space" in numerical analysis, see coarse problem.''
In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined.
The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.
Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.
==Definition==
A coarse structure on a set ''X'' is a collection E of subsets of ''X'' × ''X'' (therefore falling under the more general categorization of binary relations on ''X'') called ''controlled sets'', and so that E possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:
;1. Identity/diagonal: The diagonal Δ = is a member of E—the identity relation.
;2. Closed under taking subsets: If ''E'' is a member of E and ''F'' is a subset of ''E'', then ''F'' is a member of E.
;3. Closed under taking inverses: If ''E'' is a member of E then the inverse (or transpose) ''E'' −1 = is a member of E—the inverse relation.
;4. Closed under taking unions: If ''E'' and ''F'' are members of E then the union of ''E'' and ''F'' is a member of E.
;5. Closed under composition: If ''E'' and ''F'' are members of E then the product ''E'' o ''F'' = is a member of E—the composition of relations.
A set ''X'' endowed with a coarse structure E is a ''coarse space''.
The set ''E''() is defined as . We define the ''section'' of ''E'' by ''x'' to be the set ''E''(), also denoted ''E'' ''x''. The symbol ''E''''y'' denotes the set ''E'' −1(). These are forms of projections.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
■ウィキペディアで「:''"Coarse space" redirects here. For the use of "coarse space" in numerical analysis, see coarse problem.''In the mathematical fields of geometry and topology, a '''coarse structure''' on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined.The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.==Definition==A '''coarse structure''' on a set ''X'' is a collection '''E''' of subsets of ''X'' × ''X'' (therefore falling under the more general categorization of binary relations on ''X'') called ''controlled sets'', and so that '''E''' possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:;1. Identity/diagonal: The diagonal Δ = is a member of '''E'''—the identity relation.;2. Closed under taking subsets: If ''E'' is a member of '''E''' and ''F'' is a subset of ''E'', then ''F'' is a member of '''E'''.;3. Closed under taking inverses: If ''E'' is a member of '''E''' then the '''inverse''' (or '''transpose''') ''E'' −1 = is a member of '''E'''—the inverse relation.;4. Closed under taking unions: If ''E'' and ''F'' are members of '''E''' then the '''union''' of ''E'' and ''F'' is a member of '''E'''.;5. Closed under composition: If ''E'' and ''F'' are members of '''E''' then the '''product''' ''E'' o ''F'' = is a member of '''E'''—the composition of relations.A set ''X'' endowed with a coarse structure '''E''' is a ''coarse space''.The set ''E''() is defined as . We define the ''section'' of ''E'' by ''x'' to be the set ''E''(), also denoted ''E'' ''x''. The symbol ''E''''y'' denotes the set ''E'' −1(). These are forms of projections.」の詳細全文を読む
y'' denotes the set ''E'' −1(). These are forms of projections.
:''"Coarse space" redirects here. For the use of "coarse space" in numerical analysis, see coarse problem.''
In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined.
The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.
Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.
==Definition==
A coarse structure on a set ''X'' is a collection E of subsets of ''X'' × ''X'' (therefore falling under the more general categorization of binary relations on ''X'') called ''controlled sets'', and so that E possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:
;1. Identity/diagonal: The diagonal Δ = is a member of E—the identity relation.
;2. Closed under taking subsets: If ''E'' is a member of E and ''F'' is a subset of ''E'', then ''F'' is a member of E.
;3. Closed under taking inverses: If ''E'' is a member of E then the inverse (or transpose) ''E'' −1 = is a member of E—the inverse relation.
;4. Closed under taking unions: If ''E'' and ''F'' are members of E then the union of ''E'' and ''F'' is a member of E.
;5. Closed under composition: If ''E'' and ''F'' are members of E then the product ''E'' o ''F'' = is a member of E—the composition of relations.
A set ''X'' endowed with a coarse structure E is a ''coarse space''.
The set ''E''() is defined as . We define the ''section'' of ''E'' by ''x'' to be the set ''E''(), also denoted ''E'' ''x''. The symbol ''E''''y'' denotes the set ''E'' −1(). These are forms of projections.
抄文引用元・出典: フリー百科事典『 coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined.The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.==Definition==A coarse structure on a set ''X'' is a collection E of subsets of ''X'' × ''X'' (therefore falling under the more general categorization of binary relations on ''X'') called ''controlled sets'', and so that E possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:;1. Identity/diagonal: The diagonal Δ = is a member of E—the identity relation.;2. Closed under taking subsets: If ''E'' is a member of E and ''F'' is a subset of ''E'', then ''F'' is a member of E.;3. Closed under taking inverses: If ''E'' is a member of E then the inverse (or transpose) ''E'' −1 = is a member of E—the inverse relation.;4. Closed under taking unions: If ''E'' and ''F'' are members of E then the union of ''E'' and ''F'' is a member of E.;5. Closed under composition: If ''E'' and ''F'' are members of E then the product ''E'' o ''F'' = is a member of E—the composition of relations.A set ''X'' endowed with a coarse structure E is a ''coarse space''.The set ''E''() is defined as . We define the ''section'' of ''E'' by ''x'' to be the set ''E''(), also denoted ''E'' ''x''. The symbol ''E'y'' denotes the set ''E'' −1(). These are forms of projections.">ウィキペディア（Wikipedia）』
■ウィキペディアで「:''"Coarse space" redirects here. For the use of "coarse space" in numerical analysis, see coarse problem.''In the mathematical fields of geometry and topology, a '''coarse structure''' on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined.The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.==Definition==A '''coarse structure''' on a set ''X'' is a collection '''E''' of subsets of ''X'' × ''X'' (therefore falling under the more general categorization of binary relations on ''X'') called ''controlled sets'', and so that '''E''' possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:;1. Identity/diagonal: The diagonal Δ = is a member of '''E'''—the identity relation.;2. Closed under taking subsets: If ''E'' is a member of '''E''' and ''F'' is a subset of ''E'', then ''F'' is a member of '''E'''.;3. Closed under taking inverses: If ''E'' is a member of '''E''' then the '''inverse''' (or '''transpose''') ''E'' −1 = is a member of '''E'''—the inverse relation.;4. Closed under taking unions: If ''E'' and ''F'' are members of '''E''' then the '''union''' of ''E'' and ''F'' is a member of '''E'''.;5. Closed under composition: If ''E'' and ''F'' are members of '''E''' then the '''product''' ''E'' o ''F'' = is a member of '''E'''—the composition of relations.A set ''X'' endowed with a coarse structure '''E''' is a ''coarse space''.The set ''E''() is defined as . We define the ''section'' of ''E'' by ''x'' to be the set ''E''(), also denoted ''E'' ''x''. The symbol ''E''''y'' denotes the set ''E'' −1(). These are forms of projections.」の詳細全文を読む
y'' denotes the set ''E'' −1(). These are forms of projections.">ウィキペディア（Wikipedia）』
■coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined.The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.==Definition==A coarse structure on a set ''X'' is a collection E of subsets of ''X'' × ''X'' (therefore falling under the more general categorization of binary relations on ''X'') called ''controlled sets'', and so that E possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:;1. Identity/diagonal: The diagonal Δ = is a member of E—the identity relation.;2. Closed under taking subsets: If ''E'' is a member of E and ''F'' is a subset of ''E'', then ''F'' is a member of E.;3. Closed under taking inverses: If ''E'' is a member of E then the inverse (or transpose) ''E'' −1 = is a member of E—the inverse relation.;4. Closed under taking unions: If ''E'' and ''F'' are members of E then the union of ''E'' and ''F'' is a member of E.;5. Closed under composition: If ''E'' and ''F'' are members of E then the product ''E'' o ''F'' = is a member of E—the composition of relations.A set ''X'' endowed with a coarse structure E is a ''coarse space''.The set ''E''() is defined as . We define the ''section'' of ''E'' by ''x'' to be the set ''E''(), also denoted ''E'' ''x''. The symbol ''E'y'' denotes the set ''E'' −1(). These are forms of projections.">ウィキペディアで「:''"Coarse space" redirects here. For the use of "coarse space" in numerical analysis, see coarse problem.''In the mathematical fields of geometry and topology, a '''coarse structure''' on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined.The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.==Definition==A '''coarse structure''' on a set ''X'' is a collection '''E''' of subsets of ''X'' × ''X'' (therefore falling under the more general categorization of binary relations on ''X'') called ''controlled sets'', and so that '''E''' possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:;1. Identity/diagonal: The diagonal Δ = is a member of '''E'''—the identity relation.;2. Closed under taking subsets: If ''E'' is a member of '''E''' and ''F'' is a subset of ''E'', then ''F'' is a member of '''E'''.;3. Closed under taking inverses: If ''E'' is a member of '''E''' then the '''inverse''' (or '''transpose''') ''E'' −1 = is a member of '''E'''—the inverse relation.;4. Closed under taking unions: If ''E'' and ''F'' are members of '''E''' then the '''union''' of ''E'' and ''F'' is a member of '''E'''.;5. Closed under composition: If ''E'' and ''F'' are members of '''E''' then the '''product''' ''E'' o ''F'' = is a member of '''E'''—the composition of relations.A set ''X'' endowed with a coarse structure '''E''' is a ''coarse space''.The set ''E''() is defined as . We define the ''section'' of ''E'' by ''x'' to be the set ''E''(), also denoted ''E'' ''x''. The symbol ''E''''y'' denotes the set ''E'' −1(). These are forms of projections.」の詳細全文を読む
y'' denotes the set ''E'' −1(). These are forms of projections.">ウィキペディアで「:''"Coarse space" redirects here. For the use of "coarse space" in numerical analysis, see coarse problem.''In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined.The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.==Definition==A coarse structure on a set ''X'' is a collection E of subsets of ''X'' × ''X'' (therefore falling under the more general categorization of binary relations on ''X'') called ''controlled sets'', and so that E possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:;1. Identity/diagonal: The diagonal Δ = is a member of E—the identity relation.;2. Closed under taking subsets: If ''E'' is a member of E and ''F'' is a subset of ''E'', then ''F'' is a member of E.;3. Closed under taking inverses: If ''E'' is a member of E then the inverse (or transpose) ''E'' −1 = is a member of E—the inverse relation.;4. Closed under taking unions: If ''E'' and ''F'' are members of E then the union of ''E'' and ''F'' is a member of E.;5. Closed under composition: If ''E'' and ''F'' are members of E then the product ''E'' o ''F'' = is a member of E—the composition of relations.A set ''X'' endowed with a coarse structure E is a ''coarse space''.The set ''E''() is defined as . We define the ''section'' of ''E'' by ''x'' to be the set ''E''(), also denoted ''E'' ''x''. The symbol ''E'y'' denotes the set ''E'' −1(). These are forms of projections.」の詳細全文を読む
y'' denotes the set ''E'' −1(). These are forms of projections.」の詳細全文を読む
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