Projective hierarchy

In the mathematical field of descriptive set theory, a subset of a Polish space is projective if it is for some positive integer . Here is

if is analytic

if the complement of , , is

if there is a Polish space and a subset such that is the projection of ; that is,

The choice of the Polish space in the third clause above is not very important; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line.

Relationship to the analytical hierarchy

There is a close relationship between the relativized analytical hierarchy on subsets of Baire space (denoted by lightface letters and ) and the projective hierarchy on subsets of Baire space (denoted by boldface letters and ). Not every subset of Baire space is . It is true, however, that if a subset X of Baire space is then there is a set of natural numbers A such that X is . A similar statement holds for sets. Thus the sets classified by the projective hierarchy are exactly the sets classified by the relativized version of the analytical hierarchy. This relationship is important in effective descriptive set theory.

Projective set (disambiguation)

Projective Set may refer to:

Projective Set (game)

Projective Set (sometimes shortened to ProSet) is a real-time card game derived from the older game Set. The deck contains cards consisting of colored dots; some cards are laid out on the table and players attempt to find "Sets" among them. The word projective comes from the game's relation to Projective spaces over the finite field with two elements.

Projective Set has been studied mathematically as well as played recreationally. It has been a popular game at Canada/USA Mathcamp.

Rules

A Projective Set card has six binary attributes, or bits, generally represented by colored dots. For each color of dot, each card either has that dot or does not. There is one card for each possible combination of dots except the combination of no dots at all, making cards total.

Three cards are said to form a "set" if the total number of dots of each color is either 0 or 2. Similarly, four or more cards form a "set" if the number of dots of each color is an even number.

A card and itself could be said to form a two-card set, but as the cards in the deck are all distinct, this does not arise in actual gameplay.