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Topological Semigroup
In mathematics, a topological semigroup is a semigroup that is simultaneously a topological space, and whose semigroup operation is continuous.Artur Hideyuki TomitaOn sequentially compact both-sides cancellative semigroups with sequentially continuous addition./ref> Every topological group is a topological semigroup. See also * * * * * * * * * * * * * * * References Topological algebra Topological groups {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that 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 . Up 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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis. Formal definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
TODO
Todo may refer to: * Todo Bichig, Kalmyk ‘Clear Script’ * To-do list, a time management implementation * TODO (tag), a computer programming comment tag * ''Todo'' (album) Tōdō may refer to: * Tōkyūjutsu () or Tōdō (), a Japanese divination (fortune telling) method * Tōdōza () or Tōdō (), a Japanese guild for blind male musicians * Tōdō Heisuke (, 1844–1867), samurai * Tōdō Takatora (, 1556–1630), daimyō * Tōdō Takayuki (, 1813–1895), daimyō * Izumi Todo (), pseudonym for the staff at Toei Animation See also * To do Time management is the process of planning and exercising conscious control of time spent on specific activities, especially to increase effectiveness, efficiency, and productivity. It involves of various demands upon a person relating to work, s ... * Toto (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Algebra
In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense. Definition A topological algebra A over a topological field K is a topological vector space together with a bilinear multiplication :\cdot: A \times A \to A, :(a,b) \mapsto a \cdot b that turns A into an algebra over K and is continuous in some definite sense. Usually the ''continuity of the multiplication'' is expressed by one of the following (non-equivalent) requirements: * ''joint continuity'': for each neighbourhood of zero U\subseteq A there are neighbourhoods of zero V\subseteq A and W\subseteq A such that V \cdot W\subseteq U (in other words, this condition means that the multiplication is continuous as a map between topological spaces or * ''stereotype continuity'': for each totally bounded set S\subseteq A and for each neighbourhood of zero U\subseteq A there is a neighbourhood of zero V\s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
