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Filters In Topology
Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters, which are special families of subsets of some given set, also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called have many useful technical properties and they may often be used in place of arbitrary filters. Filters have generalizations called (also known as ) and , all of which appear naturally and repeatedly throughout topology. Examples include neighborhood filters/ bases/subbases and uniformities. Every filter is a prefilter and both are filter subbases. Every prefilter and filter subbase is contained in a unique smallest filter, which they are said to . This establishes a relationship between filters and prefilters that may often be ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filter Vs Ultrafilter
Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that performs some operation on a multimedia stream * Email filtering, the processing of email to organize it according to specified criteria * Content-control software also known as an Internet filter * Wordfilter, a script typically used on Internet forums or chat rooms * Berkeley Packet Filter, filter expression used in the qualification of network data * DSL filter, a low-pass filter installed between analog devices and a telephone line * Helicon Filter, a raster graphics editor * Filter (large eddy simulation), a mathematical operation intended to remove a range of small scales from the solution to the Navier-Stokes equations * Kalman filter, an approximating algorithm in optimal control applications and problems Device * Filter (chem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Cartan
Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of composer , physicist and mathematician , and the son-in-law of physicist Pierre Weiss. Life According to his own words, Henri Cartan was interested in mathematics at a very young age, without being influenced by his family. He moved to Paris with his family after his father's appointment at Sorbonne in 1909 and he attended secondary school at Lycée Hoche in Versailles. available also at In 1923 he started studying mathematics at École Normale Supérieure, receiving an agrégation in 1926 and a doctorate in 1928. His PhD thesis, entitled ''Sur les systèmes de fonctions holomorphes a variétés linéaires lacunaires et leurs applications'', was supervised by Paul Montel. Cartan taught at Lycée Malherbe in Caen from 1928 to 1929, at Un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subnets
A subnetwork or subnet is a logical subdivision of an IP network. Updated by RFC 6918. The practice of dividing a network into two or more networks is called subnetting. Computers that belong to the same subnet are addressed with an identical most-significant bit-group in their IP addresses. This results in the logical division of an IP address into two fields: the ''network number'' or ''routing prefix'' and the ''rest field'' or ''host identifier''. The ''rest field'' is an identifier for a specific host or network interface. The ''routing prefix'' may be expressed in Classless Inter-Domain Routing (CIDR) notation written as the first address of a network, followed by a slash character (''/''), and ending with the bit-length of the prefix. For example, is the prefix of the Internet Protocol version 4 network starting at the given address, having 24 bits allocated for the network prefix, and the remaining 8 bits reserved for host addressing. Addresses in the range to belon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subnet (mathematics)
In topology and related areas of mathematics, a subnet is a generalization of the concept of subsequence to the case of nets. The analogue of "subsequence" for nets is the notion of a "subnet". The definition is not completely straightforward, but is designed to allow as many theorems about subsequences to generalize to nets as possible. There are three non-equivalent definitions of "subnet". The first definition of a subnet was introduced by John L. Kelley in 1955 and later, Stephen Willard introduced his own (non-equivalent) variant of Kelley's definition in 1970. Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet" but they are each equivalent to the concept of "subordinate filter", which is the analog of "subsequence" for filters (they are not equivalent in the sense that there exist subordinate filters on X = \N whose filter/subordinate–filter relationship cannot be described in terms of the corresponding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood Characterization Of Topological Spaces
In the mathematical field of topology, a topological space is usually defined by declaring its open sets. However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a topological space determines a class of closed sets, of closure and interior operators, and of convergence of various types of objects. Each of these can instead be taken as the primary class of objects, with all of the others (including the class of open sets) directly determined from that new starting point. For example, in Kazimierz Kuratowski's well-known textbook on point-set topology, a topological space is defined as a set together with a certain type of "closure operator," and all other concepts are derived therefrom. Likewise, the neighborhood-based axioms (in the context of Hausdorff spaces) can be retraced to Felix Hausdorff's original definition of a topological space in Grundzüge der Mengenlehre. Many different textbooks use m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Topological Spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces. As a concrete category Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor :''U'' : Top → Set to the category of sets which assigns to each topological spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has an upper bound. In other words, for any a and b in A there must exist c in A with a \leq c and b \leq c. A directed set's preorder is called a . The notion defined above is sometimes called an . A is defined analogously, meaning that every pair of elements is bounded below. Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other authors call a set directed if and only if it is directed both upward and downward. Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast ordered sets, which need not be directed). Join-semilattices (which are partially ordered sets) are directed sets as well, but not conversely. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
