compact

compact disc (CD)

Molded plastic disc containing digital data that is scanned by a laser beam for the reproduction of recorded sound or other information. Since its commercial introduction in 1982, the audio CD has become the dominant format for high-fidelity recorded music. Digital audio data can be converted to analog form to reproduce the original audio signal (see digital-to-analog conversion). Coinvented by Philips Electronics and Sony Corp. in 1980, the compact disc has expanded beyond audio recordings into other storage-and-distribution uses, notably for computers (CD-ROM) and entertainment systems (videodisc and DVD). An audio CD can store just over an hour of music. A CD-ROM can contain up to 680 megabytes of computer data. A DVD, the same size as traditional CDs, is able to store up to 17 gigabytes of data, such as high-definition digital video files.

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(1620) Document signed by 41 male passengers on the Mayflower before landing at Plymouth (Massachusetts). Concerned that some members might leave to form their own colonies, William Bradford and others drafted the compact to bind the group into a political body and pledge members to abide by any laws that would be established. The document adapted a church covenant to a civil situation and was the basis of the colony's government.

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In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis.

Definition

Let X and Y be two topological spaces, and let C(X, Y) denote the set of all continuous maps between X and Y. Given a compact subset K of X and an open subset U of Y, let V(K, U) denote the set of all functions f in C(X, Y) such that f(K) is contained in U. Then the collection of all such V(K, U) is a subbase for the compact-open topology. (This collection does not always form a base for a topology on C(X, Y).)

Properties

  • If * is a one-point space then one can identify C(*, X) with X, and under this identification the compact-open topology agrees with the topology on X
  • If Y is T0, T1, Hausdorff, or regular, then the compact-open topology has the corresponding separation axiom.
  • If X is Hausdorff and S is a subbase for Y, then the collection {V(K, U) : U in S} is a subbase for the compact-open topology on C(X, Y).
  • If Y is a uniform space (in particular, if Y is a metric space), then the compact-open topology is equal to the topology of compact convergence. In other words, if Y is a uniform space, then a sequence {fn} converges to f in the compact-open topology if and only if for every compact subset K of X, {fn} converges uniformly to f on K. In particular, if X is compact and Y is a uniform space, then the compact-open topology is equal to the topology of uniform convergence.
  • If X, Y and Z are topological spaces, with Y locally compact Hausdorff (or even just preregular), then the composition map C(Y, Z) × C(X, Y) → C(X, Z), given by (f, g) mapsto fog, is continuous (here all the function spaces are given the compact-open topology and C(Y, Z) × C(X, Y) is given the product topology).
  • If Y is a locally compact Hausdorff (or preregular) space, then the evaluation map e : C(YZ) × Y → Z, defined by e(f, x) = f(x), is continuous. This can be seen as a special case of the above where X is a one-point space.
  • If X is compact, and if Y is a metric space with metric d, then the compact-open topology on C(X, Y) is metrisable, and a metric for it is given by e(f, g) = sup{d(f(x), g(x)) : x in X}, for f, g in C(X, Y).

References

  • Dugundji, James (1966). Topology. Boston, Massachusetts: Allyn and Bacon. ISBN B000-KWE22-K.
  • O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev; Textbook in Problems on Elementary Topology; online version

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