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Selection Principle
In mathematics, a selection principle is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of Set (mathematics), sets. The theory of selection principles studies these principles and their relations to other mathematical properties. Selection principles mainly describe covering properties, measure theory, measure- and category-theoretic properties, and local properties in topological spaces, especially function spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property. The main selection principles In 1924, Karl Menger Reprinted in ''Selecta Mathematica I'' (2002), , , pp. 155-178. introduced the following basis property for metric spaces: Every basis (topology), basis of the topology contains a sequence of sets with vanishing diameters that covers the space. Soon thereafter, Witold Hurewicz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anthropic Principle
In cosmology, the anthropic principle, also known as the observation selection effect, is the proposition that the range of possible observations that could be made about the universe is limited by the fact that observations are only possible in the type of universe that is capable of developing observers in the first place. Proponents of the anthropic principle argue that it explains why the universe has the age and the fundamental physical constants necessary to accommodate intelligent life. If either had been significantly different, no one would have been around to make observations. Anthropic reasoning has been used to address the question as to why certain measured physical constants take the values that they do, rather than some other arbitrary values, and to explain a perception that the universe appears to be finely tuned for the existence of life. There are many different formulations of the anthropic principle. Philosopher Nick Bostrom counts thirty, but the underlying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boaz Tsaban
Boaz Tsaban (Hebrew: בועז צבאן; born February 1973) is an Israeli mathematician on the faculty of Bar-Ilan University. His research interests include selection principles within set theory and nonabelian cryptology, within mathematical cryptology. Biography Boaz Tsaban grew up in Or Yehuda, a city near Tel Aviv. At the age of 16 he was selected with other high school students to attend the first cycle of a special preparation program in mathematics, at Bar-Ilan University, being admitted to regular mathematics courses at the University a year later. He completed his B.Sc., M.Sc. and Ph.D. degrees with highest distinctions. Two years as a post-doctoral fellow at Hebrew University were followed by a three-year Koshland Fellowship at the Weizmann Institute of Science before he joined the Department of Mathematics, Bar-Ilan University in 2007. Academic career In the field of selection principles, Tsaban devised the method of omission of intervals for establishing covering ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^=\aleph_1, or even shorter with beth numbers: \beth_1 = \aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term '' continuum'' for the real numbers. History Cantor believed the continuum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
σ-compact Space
In mathematics, a topological space is said to be ''σ''-compact if it is the union of countably many compact subspaces. A space is said to be ''σ''-locally compact if it is both ''σ''-compact and (weakly) locally compact. That terminology can be somewhat confusing as it does not fit the usual pattern of σ-(property) meaning a countable union of spaces satisfying (property); that's why such spaces are more commonly referred to explicitly as ''σ-compact (weakly) locally compact'', which is also equivalent to being exhaustible by compact sets. Properties and examples * Every compact space is ''σ''-compact, and every ''σ''-compact space is Lindelöf (i.e. every open cover has a countable subcover). The reverse implications do not hold, for example, standard Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second-countable Space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mathcal = \_^ of open subsets of T such that any open subset of T can be written as a union of elements of some subfamily of \mathcal. A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open subsets that a space can have. Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (R''n'') with its usual topology is second-countable. Although the usual base of open balls is uncountable, one can restrict this to the collection of all open balls with rational radii and whose centers have rational coordinates. This restricted collection is countable and still forms a basis. Properties ... [...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] |
Countable Tightness
In mathematics, a topological space X is called countably generated if the topology of X is determined by the countable sets in a similar way as the topology of a sequential space (or a Fréchet space) is determined by the convergent sequences. The countably generated spaces are precisely the spaces having countable tightness—therefore the name is used as well. Definition A topological space X is called if the topology on X is coherent with the family of its countable subspaces. In other words, any subset V \subseteq X is closed in X whenever for each countable subspace U of X the set V \cap U is closed in U; or equivalently, any subset V \subseteq X is open in X whenever for each countable subspace U of X the set V \cap U is open in U. Equivalently, X is countably generated if and only if the closure of any A \subseteq X equals the union of the closures of all countable subsets of A. Countable fan tightness A topological space X has if for every point x \in X and every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fréchet–Urysohn Space
In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X. Fréchet–Urysohn spaces are a special type of sequential space. The property is named after Maurice Fréchet and Pavel Urysohn. Definitions Let (X, \tau) be a topological space. The of S in (X, \tau) is the set: \begin \operatorname S :&= S := \left\ \end where \operatorname_X S or \operatorname_ S may be written if clarity is needed. A topological space (X, \tau) is said to be a if \operatorname_X S = \operatorname_X S for every subset S \subseteq X, where \operatorname_X S denotes the closure of S in (X, \tau). Sequentially open/closed sets Suppose that S \subseteq X is any subset of X. A sequence x_1, x_2, \ldots is if there exists a positive integer N such that x_i \in S for all indices i \geq N. The set S is called if every sequence \left(x_i\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindelöf Space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' subcover. A is a topological space such that every subspace of it is Lindelöf. Such a space is sometimes called strongly Lindelöf, but confusingly that terminology is sometimes used with an altogether different meaning. The term ''hereditarily Lindelöf'' is more common and unambiguous. Lindelöf spaces are named after the Finnish mathematician Ernst Leonard Lindelöf. Properties of Lindelöf spaces * Every compact space, and more generally every σ-compact space, is Lindelöf. In particular, every countable space is Lindelöf. * A Lindelöf space is compact if and only if it is countably compact. * Every second-countable space is Lindelöf, but not conversely. For example, there are many compact spaces that are not second-counta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
γ-space
In mathematics, a \gamma-space (gamma space) is a topological space that satisfies a certain basic selection principle. An infinite cover of a topological space is an \omega-cover if every finite subset of this space is contained in some member of the cover, and the whole space is not a member the cover. A cover of a topological space is a \gamma-cover if every point of this space belongs to all but finitely many members of this cover. A \gamma-space is a space in which every open \omega-cover contains a \gamma-cover. History Gerlits and Nagy introduced the notion of γ-spaces. They listed some topological properties and enumerated them by Greek letters. The above property was the third one on this list, and therefore it is called the γ-property. Characterizations Combinatorial characterization Let mathbb\infty be the set of all infinite subsets of the set of natural numbers. A set A\subset mathbb\infty is centered if the intersection of finitely many elements of A is inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
