In mathematics, descriptive set theory is the study of certain classes of "well-behaved" sets of real numbers, e.g. Borel sets, analytic sets, and projective sets. A major aim of descriptive set theory is to describe all of the "naturally occurring" sets of real numbers by using various constructions to build a strict hierarchy beginning with the open sets (generated by the open intervals). Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... This article is about sets in mathematics. ... The text or formatting below is generated by a template which has been proposed for deletion. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ... In mathematical logic and descriptive set theory, the analytical hierarchy is a second-order analogue of the arithmetical hierarchy. ... In mathematical logic and descriptive set theory, the analytical hierarchy is a second-order analogue of the arithmetical hierarchy. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

More generally, Polish spaces are studied in descriptive set theory; as it turns out, every Polish space is homeomorphic to a subspace of the Hilbert cube. This is a glossary of some terms used in the branch of mathematics known as topology. ... This word should not be confused with homomorphism. ... In mathematics, if a set with certain properties is called a space, then a subset of which with same properties is usually called a subspace. ... In mathematics, the Hilbert cube is a topological space that provides an instructive example of some ideas in topology. ...

Many questions in descriptive set theory ultimately depend upon set-theoretic considerations and the properties of ordinal and cardinal numbers. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...

References

• A. Kechris, Classical Descriptive Set Theory, GTM 156, Springer-Verlag, 1995.
• Y. Moschovakis, Descriptive Set Theory, North-Holland, 1980.