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Fine Uniformity
In mathematics, a topological space ''X'' is uniformizable if there exists a uniform structure on ''X'' that induces the topology of ''X''. Equivalently, ''X'' is uniformizable if and only if it is homeomorphic to a uniform space (equipped with the topology induced by the uniform structure). Any (pseudo)metrizable space is uniformizable since the (pseudo)metric uniformity induces the (pseudo)metric topology. The converse fails: There are uniformizable spaces that are not (pseudo)metrizable. However, it is true that the topology of a uniformizable space can always be induced by a ''family'' of pseudometrics; indeed, this is because any uniformity on a set ''X'' can be defined by a family of pseudometrics. Showing that a space is uniformizable is much simpler than showing it is metrizable. In fact, uniformizability is equivalent to a common separation axiom: :''A topological space is uniformizable if and only if it is completely regular.'' Induced uniformity One way to constru ... [...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] |
Initial Uniformity
In a written or published work, an initial is a letter at the beginning of a word, a chapter, or a paragraph that is larger than the rest of the text. The word is ultimately derived from the Latin ''initiālis'', which means ''of the beginning''. An initial is often several lines in height, and, in older books or manuscripts, may take the form of an inhabited or historiated initial. There are certain important initials, such as the Beatus initial, or B, of ''Beatus vir...'' at the opening of Psalm 1 at the start of a vulgate Latin. These specific initials in an illuminated manuscript were also called initia (singular: initium). History The classical tradition was slow to use capital letters for initials at all; in surviving Roman texts it often is difficult even to separate the words as spacing was not used either. In late antiquity (–6th century) both came into common use in Italy, the initials usually were set in the left margin (as in the second example below), as thoug ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which Comparison of topologies, agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a Product (category theory), categorical product of its factors, whereas the box topology is too Comparison of topologies, fine; in that sense the product topology is the natural topology on the Cartesian product. Definition Throughout, I will be some non-empty index set and for every index i \in I, let X_i be a topological space. Denote the Cartesian product of the sets X_i by X := \prod X_ := \prod_ X_i and for every index i \in I, denote the i-th by \begin p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forgetful Functor
In mathematics, more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's structure or properties mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case. Overview As an example, there are several forgetful functors from the category of commutative rings. A ( unital) ring, described in the language of universal algebra, is an ordered tuple (R,+,\times,a,0,1) satisfying certain axioms, where + and \times are binary functions on the set R, a is a unary operation corresponding to additive inverse, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Left Adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects c in \mathcal and d in \mathcal, a bijection between the respective morphism sets :\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each Mathematical object, object X in ''C'' to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field (mathematics), field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of category (mathematics), categories and functors by means of a universal morphism (see , below). Universal morphisms can also be thought more abstractly as Initia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coarser Topology
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the collection of subsets which are considered to be "open". (An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.) For definiteness the reader should think of a topology as the family of open sets of a topological space, since that is the standard meaning of the word "topology". Let ''τ''1 and ''τ''2 be two topologies on a set ''X'' such that ''τ''1 is contained in ''τ''2: :\tau_1 \subseteq \tau_2. That is, every element of ''τ''1 is also an element of ''τ''2. Then the topology ''τ''1 is said to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Topology
In general topology and related areas of mathematics, the initial topology (or induced topology or strong topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on X that makes those functions continuous. The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these. The dual notion is the final topology, which for a given family of functions mapping to a set Y is the finest topology on Y that makes those functions continuous. Definition Given a set X and an indexed family \left(Y_i\right)_ of topological spaces with functions f_i : X \to Y_i, the initial topology \tau on X is the coarsest topology on X such that each f_i : (X, \tau) \to Y_i is continuous. Definition in terms of open sets If \left(\tau_i\right)_ is a family of topologies X indexed by I \neq \varnothing, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entourage (topology)
In the mathematical field of topology, a uniform space is a set with additional structure that is used to define '' uniform properties'', such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis. In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "''x'' is closer to ''a'' than ''y'' is to ''b''" make sense in uniform spaces. By comparison, in a general topological space, given sets ''A,B'' it is meaningful to say that a point ''x'' is ''arbitrarily close'' to ''A'' (i.e., in the closure of ''A''), or perhaps that ''A'' is a ''smaller neighborhood'' of ''x'' than ''B'', but notions of closeness of points and relative closeness are not described well by topological structure alone. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformly Continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number \varepsilon, then there is a positive real number \delta such that , f(x) - f(y), 0 there exists a real number \delta > 0 such that for every x,y \in X with d_1(x,y) 0 such that for every x,y \in X , , x - y, 0 \; \forall x \in X \; \forall y \in X : \, d_1(x,y) 0 , \forall x \in X , and \forall y \in X ) are used. * Equivalently, f is uniformly continuous if it admits a modulus of continuity. Definition of (ordinary) continuity * f is called continuous \underline if for every real number \varepsilon > 0 there exists a real number \delta > 0 such that for every y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
