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λ-Suslin
In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More generally, a subset ''A'' of ''κ''ω is ''λ''-Suslin if there is a tree ''T'' on ''κ'' × ''λ'' such that ''A'' = p 'T'' By a tree on ''κ'' × ''λ'' we mean a subset ''T'' ⊆ ⋃''n''<ω(''κ''''n'' × ''λ''''n'') closed under initial segments, and p 'T''= is the projection of ''T'', where 'T''= is the set of es through ''T''. Since 'T''is a closed set for the |
Suslin Cardinal
In mathematics, a cardinal λ < Θ is a Suslin cardinal if there exists a set P ⊂ 2ω such that P is λ-Suslin but P is not λ'-Suslin for any λ' < λ. It is named after the n Mikhail Yakovlevich Suslin (1894–1919). See also *Suslin representation
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Space (set Theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology, called the product topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called "reals". It is denoted by \N^, or ωω, or by the symbol \mathcal or sometimes by ωω (not to be confused with the countable ordinal obtained by ordinal exponentiation). The Baire space is defined to be the Cartesian product of countably infinitely many copies of the set of natural numbers, and is given the product topology (where each copy of the set of natural numbers is given the discrete topology). The Baire space is often represented using the tree of finite sequences of natural numbers. (This space should also not be confused with the concept of a Baire space, which is a certain kind of topological space.) The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits. Topology and trees ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (descriptive Set Theory)
In descriptive set theory, a tree on a set X is a collection of finite sequences of elements of X such that every prefix of a sequence in the collection also belongs to the collection. Definitions Trees The collection of all finite sequences of elements of a set X is denoted X^. With this notation, a tree is a nonempty subset T of X^, such that if \langle x_0,x_1,\ldots,x_\rangle is a sequence of length n in T, and if 0\le m and called the ''body'' of the tree T. A tree that has no branches is called '' wellfounded''; a tree with at least one branch is ''illfounded''. By Kőnig's lemma, a tree on a finite set with an infinite number of sequences must necessarily be illfounded. Terminal nodes A finite sequence that belongs to a tree T is called a terminal node if it is not a prefix of a longer sequence in T. Equivalently, \langle x_0,x_1,\ldots,x_\rangle \in T is terminal if there is no element x of X such that that \langle x_0,x_1,\ldots,x_,x\rangle \in T. A tree that does no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which Comparison of topologies, agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a Product (category theory), categorical product of its factors, whereas the box topology is too Comparison of topologies, fine; in that sense the product topology is the natural topology on the Cartesian product. Definition Throughout, I will be some non-empty index set and for every index i \in I, let X_i be a topological space. Denote the Cartesian product of the sets X_i by X := \prod X_ := \prod_ X_i and for every index i \in I, denote the i-th by \begin p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Definitions Given a set X: A metric space (E,d) is said to be '' uniformly discrete'' if there exists a ' r > 0 such that, for any x,y \in E, one has either x = y or d(x,y) > r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set \left\. Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Set Theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and Group action (mathematics), group actions, and mathematical logic. Polish spaces Descriptive set theory begins with the study of Polish spaces and their Borel sets. A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line \mathbb, the Baire space (set theory), Baire space \mathcal, the Cantor space \mathcal, and the Hilbert cube I^. Universality properties The class of Polish spaces has several universality properties, which show that there is no loss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Operation
In mathematics, the Suslin operation 𝓐 is an operation that constructs a set from a collection of sets indexed by finite sequences of positive integers. The Suslin operation was introduced by and . In Russia it is sometimes called the A-operation after Alexandrov. It is usually denoted by the symbol 𝓐 (a calligraphic capital letter A). Definitions A Suslin scheme is a family P = \ of subsets of a set X indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set :\mathcal A P = \bigcup_ \bigcap_ P_ Alternatively, suppose we have a Suslin scheme, in other words a function M from finite sequences of positive integers n_1,\dots, n_k to sets M_. The result of the Suslin operation is the set : \mathcal A(M) = \bigcup \left(M_ \cap M_ \cap M_ \cap \dots \right) where the union is taken over all infinite sequences n_1,\dots, n_k, \dots If M is a family of subsets of a set X, then \mathcal A(M) is the family of subsets of X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
