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Axiom Of Dependent Choice
In 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 ar ..., the axiom of dependent choice, denoted by \mathsf , is a weak form of the axiom of choice ( \mathsf ) that is still sufficient to develop much of real analysis. It was introduced by Paul Bernays in a 1942 article in reverse mathematics that explores which set-theoretic axioms are needed to develop analysis."The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame." The axiom of dependent choice is stated on p. 86. Formal statement A homogeneous relation R on X is called a total relation if for every a \in X, there exists some b \in X such that a\,R~b is true. The axiom of dependent choice can be stated as follows: For every ... [...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] |
Baire Category Theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense). It is used in the proof of results in many areas of analysis and geometry, including some of the fundamental theorems of functional analysis. Versions of the Baire category theorem were first proved independently in 1897 by Osgood for the real line \R and in 1899 by Baire for Euclidean space \R^n. The more general statement for completely metrizable spaces was first shown by Hausdorff in 1914. Statement A Baire space is a topological space X in which every countable intersection of open dense sets is dense in X. See the corresponding article for a list of equivalent characterizations, as some are more useful than others depending on the application. * (BCT1) Every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Set Property
In the mathematical field of descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a perfect set. As nonempty perfect sets in a Polish space always have the cardinality of the continuum, and the reals form a Polish space, a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of reals has the cardinality of the continuum. The Cantor–Bendixson theorem states that closed sets of a Polish space ''X'' have the perfect set property in a particularly strong form: any closed subset of ''X'' can be written uniquely as the disjoint union of a perfect set and a countable set. In particular, every uncountable Polish space has the perfect set property, and can be written as the disjoint union of a perfect set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Property Of Baire
A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such that A \bigtriangleup U is meager (where \bigtriangleup denotes the symmetric difference).. Definitions A subset A \subseteq X of a topological space X is called almost open and is said to have the property of Baire or the Baire property if there is an open set U\subseteq X such that A \bigtriangleup U is a meager subset, where \bigtriangleup denotes the symmetric difference. Further, A has the Baire property in the restricted sense if for every subset E of X the intersection A\cap E has the Baire property relative to E. Properties The family of sets with the property of Baire forms a σ-algebra. That is, the complement of an almost open set is almost open, and any countable union or intersection of almost open sets is again almos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonmeasurable
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zermelo–Fraenkel set theory, the axiom of choice entails that non-measurable subsets of \mathbb exist. The notion of a non-measurable set has been a source of great controversy since its introduction. Historically, this led Borel and Kolmogorov to formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections of intervals (called Borel sets) plus-minus null sets. These sets are rich enough to include every conceivable definition of a set that arises in standard mathematics, but they require a lot of formalism to prove that sets are measurable. In 1970, Robert M. Solovay constructed the Solovay model, which shows that it is consistent with standard set theory wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strictly Increasing
In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive meaning of the term is to be understood. The opposite is non-strict, which is often understood to be the case but can be put explicitly for clarity. In some contexts, the word "proper" can also be used as a mathematical synonym for "strict". Use This term is commonly used in the context of inequalities — the phrase "strictly less than" means "less than and not equal to" (likewise "strictly greater than" means "greater than and not equal to"). More generally, a strict partial order, strict total order, and strict weak order exclude equality and equivalence. When comparing numbers to zero, the phrases "strictly positive" and "strictly negative" mean "positive and not equal to zero" and "negative and not equal to zero", respectiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Mathematical Bulletin
The ''Canadian Mathematical Bulletin'' () is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antonio Lei and Javad Mashreghi. The journal publishes short articles in all areas of mathematics that are of sufficient interest to the general mathematical public. Abstracting and indexing The journal is abstracted in: for the Canadian Mathematical Bulletin. * '''' * '' |
Chain (ordered Set)
A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A chain may consist of two or more links. Chains can be classified by their design, which can be dictated by their use: * Those designed for lifting, such as when used with a hoist; for pulling; or for securing, such as with a bicycle lock, have links that are torus-shaped, which make the chain flexible in two dimensions (the fixed third dimension being a chain's length). Small chains serving as jewellery are a mostly decorative analogue of such types. * Those designed for transferring power in machines have links designed to mesh with the teeth of the sprockets of the machine, and are flexible in only one dimension. They are known as roller chains, though there are also non-roller chains such as block chains. Two distinct chains can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-ordered
In mathematics, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty subset of has a least element in this ordering. The set together with the ordering is then called a well-ordered set (or woset). In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than . There may be elements, besides the least element, that have no predecessor (see below for an example). A well-ordered set contains for every subset with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of in . If ≤ is a non-strict well ordering, then < is a stric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relations, referred to in this article as ''non-strict'' partial orders. However som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zorn's Lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element. The lemma was proved (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935. It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure. Zorn's lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that within ZF ( Zermelo–Fraenkel set theory without th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch (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 not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
