This article lists some properties of sets of real numbers. The general study of these concepts forms descriptive set theory, which has a rather different emphasis from general topology.
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.
In the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible. A topological space T is called meagre if it is a meager subset of itself; otherwise, it is called nonmeagre.
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.
In the mathematical field of descriptive set theory, a subset of a Polish space is an analytic set if it is a continuous image of a Polish space. These sets were first defined by Luzin (1917) and his student Souslin (1917).
A subset of a topological space has the property of Baire, or is called an almost open set, if it differs from an open set by a meager set;
In set theory, L(R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals.
Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists.
In mathematics, Baire functions are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by René-Louis Baire in 1899. A Baire set is a set whose characteristic function is a Baire function.
In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets.
Mikhail Yakovlevich Suslin was a Russian mathematician who made major contributions to the fields of general topology and descriptive set theory.
In descriptive set theory, a set is said to be homogeneously Suslin if it is the projection of a homogeneous tree. is said to be -homogeneously Suslin if it is the projection of a -homogeneous tree.
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by some sort of definability property; for example, the collection of all open sets in some fixed collection of Polish spaces is a pointclass.
In mathematics, a Baire measure is a measure on the σ-algebra of Baire sets of a topological space whose value on every compact Baire set is finite. In compact metric spaces the Borel sets and the Baire sets are the same, so Baire measures are the same as Borel measures that are finite on compact sets. In general Baire sets and Borel sets need not be the same. In spaces with non-Baire Borel sets, Baire measures are used because they connect to the properties of continuous functions more directly.
This is a glossary of set theory.
In mathematics, the Kuratowski–Ulam theorem, introduced by Kazimierz Kuratowski and Stanislaw Ulam (1932), called also the Fubini theorem for category, is an analog of Fubini's theorem for arbitrary second countable Baire spaces.
In measure theory, projection maps often appear when working with product spaces: The product sigma-algebra of measurable spaces is defined to be the finest such that the projection mappings will be measurable. Sometimes for some reasons product spaces are equipped with sigma-algebra different than the product sigma-algebra. In these cases the projections need not be measurable at all.