Infinity-Borel set

In set theory, a subset of a Polish space is ∞-Borel if it can be obtained by starting with the open subsets of , and transfinitely iterating the operations of complementation and wellordered union. Note that the set of ∞-Borel sets may not actually be closed under wellordered union; see below.

Formal definition

More formally: we define by simultaneous transfinite recursion the notion of ∞-Borel code, and of the interpretation of such codes. Since is Polish, it has a countable base. Let enumerate that base (that is, is the basic open set). Now:

Every natural number is an ∞-Borel code. Its interpretation is .

If is an ∞-Borel code with interpretation , then the ordered pair is also an ∞-Borel code, and its interpretation is the complement of , that is, .

If is a length-α sequence of ∞-Borel codes for some ordinal α (that is, if for every β<α, is an ∞-Borel code, say with interpretation ), then the ordered pair is an ∞-Borel code, and its interpretation is .

Now a set is ∞-Borel if it is the interpretation of some ∞-Borel code.

Set (music)

A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.

A set by itself does not necessarily possess any additional structure, such as an ordering. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called segments); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis.

Two-element sets are called dyads, three-element sets trichords (occasionally "triads", though this is easily confused with the traditional meaning of the word triad). Sets of higher cardinalities are called tetrachords (or tetrads), pentachords (or pentads), hexachords (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g.,), octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.

Set (darts)

A set in darts consists of a sequence of legs (games) played, ending when the count of legs won meets certain criteria. Throwing first is considered an advantage in a leg, so players alternate who throws first in each leg during the set.

Generally, a set will consist of the best of five legs (first player to win three) - although there are some exceptions. The most notable being the Winmau World Masters, where a set is the best of three legs (first to two).

Example

During the final of the 2007 PDC World Darts Championship, Raymond van Barneveld defeated Phil Taylor by seven sets to six in one of the most dramatic darts matches of all-time. The breakdown of how each set went is shown here.

Taylor won the first two sets by three legs to nil, then added the third set before van Barneveld took the next two. The match continued and went to six sets all. Neither player managed to gain a two-leg advantage in the final set "tiebreak" so the deciding leg was played when the score reached 5-5. Before the final leg, the players threw for the bullseye with the nearest given the advantage of throwing first.

Set construction

Set construction is the process by which a construction manager undertakes to build full scale scenery suitable for viewing by camera, as specified by a production designer or art director working in collaboration with the director of a production to create a set for a theatrical, film or television production. The set designer produces a scale model, scale drawings, paint elevations (a scale painting supplied to the scenic painter of each element that requires painting), and research about props, textures, and so on. Scale drawings typically include a groundplan, elevation, and section of the complete set, as well as more detailed drawings of individual scenic elements which, in theatrical productions, may be static, flown, or built onto scenery wagons. Models and paint elevations are frequently hand-produced, though in recent years, many Production Designers and most commercial theatres have begun producing scale drawings with the aid of computer drafting programs such as AutoCAD or Vectorworks.