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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 sets. The theory of selection principles studies these principles and their relations to other mathematical properties. Selection principles mainly describe covering properties, 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 introduced the following basis property for metric spaces: Every basis of the topology contains a sequence of sets with vanishing diameters that covers the space. Soon thereafter, Witold Hurewicz observed that Menger's basis property is equivalent to the following selective property: for every sequence of op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anthropic Principle
The anthropic principle, also known as the "observation selection effect", is the hypothesis, first proposed in 1957 by Robert Dicke, that there is a restrictive lower bound on how statistically probable our observations of the universe are, because observations could only happen in a universe capable of developing intelligent life. Proponents of the anthropic principle argue that it explains why this universe has the age and the fundamental physical constants necessary to accommodate conscious life, since if either had been different, we would not have been around to make observations. Anthropic reasoning is often used to deal with the notion that the universe seems to be finely tuned for the existence of life. There are many different formulations of the anthropic principle. Philosopher Nick Bostrom counts them at thirty, but the underlying principles can be divided into "weak" and "strong" forms, depending on the types of cosmological claims they entail. The weak anthropic p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that 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 '' the continuum'' for the real numbers. History Cantor believed the continuum hypothes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointwise Convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and Y is a topological space, such as the real or complex numbers or a metric space, for example. A net or sequence of functions \left(f_n\right) all having the same domain X and codomain Y is said to converge pointwise to a given function f : X \to Y often written as \lim_ f_n = f\ \mbox if (and only if) \lim_ f_n(x) = f(x) \text x \text f. The function f is said to be the pointwise limit function of the \left(f_n\right). Sometimes, authors use the term bounded pointwise convergence when there is a constant C such that \forall n,x,\;, f_n(x), Properties This concept is often contrasted with |
Tychonoff Space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is also a Hausdorff space; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff). Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (Тихонов) is variously rendered as "Tychonov", "Tikhonov", "Tihonov", "Tichonov", etc. who introduced them in 1930 in order to avoid the pathological situation of Hausdorff spaces whose only continuous real-valued functions are constant maps. Definitions A topological space X is called if points can be separated from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any closed set A \subseteq X and any point x \in X \setminus A, there exists a real-valued continuous function f : X \to \R suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Measure Zero Set
In mathematical analysis, a strong measure zero set is a subset ''A'' of the real line with the following property: :for every sequence (ε''n'') of positive reals there exists a sequence (''In'') of intervals such that , ''I''''n'', < ε''n'' for all ''n'' and ''A'' is contained in the union of the ''I''''n''. (Here , ''I''''n'', denotes the length of the interval ''I''''n''.) Every is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has 0. The is an exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meagre Set
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. A subset of X is called X, a of X, or of the in X if it is a countable union of nowhere dense subsets of X (where a nowhere dense set is a set whose closure has empty interior). The qualifier "in X" can be omitted if the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact. Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces. First examples Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D-space
In mathematics, a topological space X is a D-space if for any family \ of open sets such that x\in U_x for all points x\in X, there is a closed discrete subset D of the space X such that \bigcup_U_x=X. History The notion of D-spaces was introduced by Eric Karel van Douwen and E.A. Michael. It first appeared in a 1979 paper by van Douwen and Washek Frantisek Pfeffer in the Pacific Journal of Mathematics. Whether every Lindelöf and regular topological space is a D-space is known as the D-space problem. This problem is among twenty of the most important problems of set theoretic topology. Properties * Every Menger space is a D-space. * A subspace of a topological linearly ordered space is a D-space iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicon ... it is a paracompact ... [...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. This space is commonly used in descriptive set theory, to the extent that its elements are often called "reals". It is denoted NN, ωω, by the symbol \mathcal or also 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. The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits. Topology and trees The product topology used to define the Baire space can be described more concretely in terms of trees. The basic open sets of the product topology are cylinder ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
