词条 | Coarse structure |
释义 |
"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. DefinitionA 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:
A set X endowed with a coarse structure E is a coarse space. The set E[K] is defined as {x in X : there is a y in K such that (x, y) is in E}. We define the section of E by x to be the set E[{x}], also denoted E x. The symbol Ey denotes the set E −1[{y}]. These are forms of projections. IntuitionThe controlled sets are "small" sets, or "negligible sets": a set A such that A × A is controlled is negligible, while a function f : X → X such that its graph is controlled is "close" to the identity. In the bounded coarse structure, these sets are the bounded sets, and the functions are the ones that are a finite distance from the identity in the uniform metric. Coarse MapsGiven a set S and a coarse structure X, we say that the maps and are close if is a controlled set. A subset B of X is said to be bounded if is a controlled set. For coarse structures X and Y, we say that is coarse if for each bounded set B of Y the set is bounded in X and for each controlled set E of X the set is controlled in Y.[1] X and Y are said to be coarsely equivalent if there exists coarse maps and such that is close to and is close to . Examples
With this structure, the integer lattice Zn is coarsely equivalent to n-dimensional Euclidean space.
In this structure, a map is a coarse equivalence if and only if it is a bijection (of sets).
See also
References
| last = Roe | first = John | title = What is...a Coarse Space? | journal = Notices of the American Mathematical Society |date=June–July 2006 | volume = 53 | issue = 6 | pages = 669 | url = http://www.ams.org/notices/200606/whatis-roe.pdf | format = PDF | accessdate = 2008-01-16 }} 2 : Metric geometry|Topology |
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