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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] |
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] |
Derived Set (mathematics)
In mathematics, more specifically in point-set topology, the derived set of a subset S of a topological space is the set of all limit points of S. It is usually denoted by S'. The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line. Definition The derived set of a subset S of a topological space X, denoted by S', is the set of all points x \in X that are limit points of S, that is, points x such that every neighbourhood of x contains a point of S other than x itself. Examples If \Reals is endowed with its usual Euclidean topology then the derived set of the half-open interval , 1) is the closed interval [0, 1 Consider \Reals with the Topology (structure)">topology (open sets) consisting of the empty set and any subset of \Reals that contains 1. The derived set of A := \ is A' = \Reals \setminus \. Properties Let X denote a topological space in what follows. If A and B are subsets of X, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Game
In mathematics, a topological game is an infinite game of perfect information played between two players on a topological space. Players choose objects with topological properties such as points, open sets, closed sets and open coverings. Time is generally discrete, but the plays may have Transfinite number, transfinite lengths, and extensions to continuum time have been put forth. The conditions for a player to win can involve notions like topological closure and wikt:convergence, convergence. It turns out that some fundamental topological constructions have a natural counterpart in topological games; examples of these are the Baire property, Baire spaces, completeness and convergence properties, separation properties, covering and base properties, continuous images, Suslin sets, and singular spaces. At the same time, some topological properties that arise naturally in topological games can be generalized beyond a game-theoretic context: by virtue of this duality, topological games ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thumb is ''pollex'' (compare ''hallux'' for big toe), and the corresponding adjective for thumb is ''pollical''. Definition Thumb and fingers The English word ''finger'' has two senses, even in the context of appendages of a single typical human hand: 1) Any of the five terminal members of the hand. 2) Any of the four terminal members of the hand, other than the thumb. Linguistically, it appears that the original sense was the first of these two: (also rendered as ) was, in the inferred Proto-Indo-European language, a suffixed form of (or ), which has given rise to many Indo-European-family words (tens of them defined in English dictionaries) that involve, or stem from, concepts of fiveness. The thumb shares the following with each of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Set-theoretic definition A rigorous definition of the Cartesian product re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Ordinal Number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally using linearly ordered greek letter variables that include the natural numbers and have the property that every set of ordinals has a least or "smallest" element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega (omega) to be the least element that is greater than every natural number, along with ordinal numbers , , etc., which are even greater than . A linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-orde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Set
In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is * \boldsymbol^1_1 if A is analytic * \boldsymbol^1_n if the complement of A, X\setminus A, is \boldsymbol^1_n * \boldsymbol^1_ if there is a Polish space Y and a \boldsymbol^1_n subset C\subseteq X\times Y such that A is the projection of C onto X; that is, A=\. The choice of the Polish space Y in the third clause above is not very important; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line. Relationship to the analytical hierarchy There is a close relationship between the relativized analytical hierarchy on subsets of Baire space (denoted by lightface letters \Sigma and \Pi) and the projective hierarchy on subsets of Baire space (denoted by boldface letters \boldsymbol and \boldsymbol). Not every \boldsymbol^1_n subset of Baire space is \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ωα). The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more". There is a rough convention that results provable from ZFC alone may be stated without hypotheses, but that if the proof requires other assumptions (such as the existence of large cardinals), these should be stated. Whether this is simply a linguistic convention, or something more, is a controversial point among distinct ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Set
In the mathematical field of descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student . Definition There are several equivalent definitions of analytic set. The following conditions on a subspace ''A'' of a Polish space ''X'' are equivalent: *''A'' is analytic. *''A'' is empty or a continuous image of the Baire space ωω. *''A'' is a Suslin space, in other words ''A'' is the image of a Polish space under a continuous mapping. *''A'' is the continuous image of a Borel set in a Polish space. *''A'' is a Suslin set, the image of the Suslin operation. *There is a Polish space Y and a Borel set B\subseteq X\times Y such that A is the projection of B onto X; that is, : A=\. *''A'' is the projection of a closed set in the cartesian product of ''X'' with the Baire space. *''A'' is the projection of a Gδ set in the cartesian product of ''X'' with the Cantor space 2� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo Fraenkel Set Theory
Ernst Friedrich Ferdinand Zermelo (; ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929 work on ranking chess players is the first description of a model for pairwise comparison that continues to have a profound impact on various applied fields utilizing this method. Life Ernst Zermelo graduated from Berlin's Luisenstädtisches Gymnasium (now ) in 1889. He then studied mathematics, physics and philosophy at the University of Berlin, the University of Halle, and the University of Freiburg. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations (''Untersuchungen zur Variationsrechnung''). Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, under whos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solovay's Model
In the mathematical field of set theory, the Solovay model is a model constructed by in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal. In this way Solovay showed that in the proof of the existence of a non-measurable set from ZFC (Zermelo–Fraenkel set theory plus the axiom of choice), the axiom of choice is essential, at least granted that the existence of an inaccessible cardinal is consistent with ZFC. Statement ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice. Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension ''V'' 'G''such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property. Construction Solov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
