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Baire Space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces. The Baire category theorem combined with the properties of Baire spaces has numerous applications in topology, geometry, and analysis, in particular functional analysis. For more motivation and applications, see the article Baire category theorem. The current article focuses more on characterizations and basic properties of Baire spaces per se. Bourbaki introduced the term "Baire space" in honor of René Baire, who investigated the Baire category theorem in the context of Euclidean space \R^n in his 1899 thesis. Definition The definition that follows is based on the notions of meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and nonmeagre ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Nonmeagre
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms. The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre. Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis. Definitions Throughout, X will be a topological space. The definition of meagre set uses the notion of a nowhere dense subset of X, that is, a subset of X whose closure has empty interior. See the corresponding article for more details. A subset of X is called X, a of X, or of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish Space
In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that has a countable Dense set, dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Alfred Tarski, 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. Common examples of Polish spaces are the real line, any Separable space, separable Banach space, the Cantor space, and the Baire space (set theory), Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; ... [...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] |
Homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation of a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end ... [...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] |
Locally Compact Hausdorff
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completely Metrizable
In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (''X'', ''T'') for which there exists at least one metric ''d'' on ''X'' such that (''X'', ''d'') is a complete metric space and ''d'' induces the topology ''T''. The term topologically complete space is employed by some authors as a synonym for ''completely metrizable space'', but sometimes also used for other classes of topological spaces, like completely uniformizable spaces or Čech-complete spaces. Difference between ''complete metric space'' and ''completely metrizable space'' The distinction between a ''completely metrizable space'' and a ''complete metric space'' lies in the words ''there exists at least one metric'' in the definition of completely metrizable space, which is not the same as ''there is given a metric'' (the latter would yield the definition of complete metric space). Once we make the choice of the metric on a completely metrizable space (out of al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudometric Space
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy, the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis. When a topology is generated using a family of pseudometrics, the space is called a gauge space. Definition A pseudometric space (X,d) is a set X together with a non-negative real-valued function d : X \times X \longrightarrow \R_, called a , such that for every x, y, z \in X, #d(x,x) = 0. #''Symmetry'': d(x,y) = d(y,x) #'' Subadditivity''/''Triangle inequality'': d(x,z) \leq d(x,y) + d(y,z) Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have d(x, y) = 0 for dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a Set (mathematics), set , often denoted by A^c (or ), is the set of Element (mathematics), elements not in . When all elements in the Universe (set theory), universe, i.e. all elements under consideration, are considered to be Element (mathematics), members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
