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Alexander Arhangelskii
Alexander Vladimirovich Arhangelskii (, ''Aleksandr Vladimirovich Arkhangelsky'', born 13 March 1938 in Moscow) is a Russian mathematician. His research, comprising over 200 published papers, covers various subfields of general topology. He has done particularly important work in metrizability theory and generalized metric spaces, cardinal functions, topological function spaces and other topological groups, and special classes of topological maps. After a long and distinguished career at Moscow State University, he moved to the United States in the 1990s. In 1993 he joined the faculty of Ohio University, from which he retired in 2011. Biography Arhangelskii was the son of Vladimir Alexandrovich Arhangelskii and Maria Pavlova Radimova, who divorced by the time he was four years old. He was raised in Moscow by his father. He was also close to his uncle, childless aircraft designer Alexander Arkhangelsky. In 1954, Arhangelskii entered Moscow State University, where he became a stud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moscow
Moscow is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia, standing on the Moskva (river), Moskva River in Central Russia. It has a population estimated at over 13 million residents within the city limits, over 19.1 million residents in the urban area, and over 21.5 million residents in Moscow metropolitan area, its metropolitan area. The city covers an area of , while the urban area covers , and the metropolitan area covers over . Moscow is among the world's List of largest cities, largest cities, being the List of European cities by population within city limits, most populous city entirely in Europe, the largest List of urban areas in Europe, urban and List of metropolitan areas in Europe, metropolitan area in Europe, and the largest city by land area on the European continent. First documented in 1147, Moscow became the capital of the Grand Principality of Moscow, which led the unification of the Russian lan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Arkhangelsky (aircraft Designer)
Aleksandr Aleksandrovich Arkhangelsky (, 1892 – 18 December 1978) was a Soviet and Russian aircraft designer and doctor of technical sciences. Hero of Socialist Labour (1947) Biography Arkhangelsky was born in 1892, and graduated from MVTU in 1918. During his studies, he worked at the aerodynamic laboratory headed by Nikolay Zhukovsky, then worked at TsAGI from 1918 to 1936. He designed and built several aerosleds ARBES along with B. S. Stechkin. After the establishment of the aircraft design bureau of Andrei Tupolev at TsAGI, he participated in all ANT designs. In 1932, he was appointed chief of the department of high-speed aircraft. He was the leading designer of the first Soviet bomber, the ANT-40 (SB), and its transport development, the PS-35. From 1936 on, he was chief of the bureau and responsible for large-scale production of the SB. He was the chief designer of the Ar-2. Arhangelsky OKB rejoined Tupolev OKB in 1941. In 1947, he became first deputy chief ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thumb is ''pollex'' (compare ''hallux'' for big toe), and the corresponding adjective for thumb is ''pollical''. Definition Thumb and fingers The English word ''finger'' has two senses, even in the context of appendages of a single typical human hand: 1) Any of the five terminal members of the hand. 2) Any of the four terminal members of the hand, other than the thumb. Linguistically, it appears that the original sense was the first of these two: (also rendered as ) was, in the inferred Proto-Indo-European language, a suffixed form of (or ), which has given rise to many Indo-European-family words (tens of them defined in English dictionaries) that involve, or stem from, concepts of fiveness. The thumb shares the following with each of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff ... [...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. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''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 topological spaces. One suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-countable Space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point x in X there exists a sequence N_1, N_2, \ldots of neighbourhoods of x such that for any neighbourhood N of x there exists an integer i with N_i contained in N. Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods. Examples and counterexamples The majority of 'everyday' spaces in mathematics are first-countable. In particular, every metric space is first-countable. To see this, note that the set of open balls centered at x with radius 1/n for integers form a countable local base at x. An example of a space that is not first-countable is the cofinite topology on an uncountable s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Urysohn
Pavel Samuilovich Urysohn (in Russian language, Russian: ; 3 February, 1898 – 17 August, 1924) was a Soviet Union, Soviet mathematician who is best known for his contributions in Dimension theory (algebra), dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are fundamental results in topology. He also constructed what is now called the Urysohn universal space and his name is also commemorated in the terms Fréchet–Urysohn space, Menger–Urysohn dimension and Urysohn integral equation. He and Pavel Alexandrov formulated the modern definition of Compact space, compactness in 1923. Biography Pavel Urysohn was born in Odesa, Odesa in 1898. His mother died when he was little, and he entered the care of his father and sister. The family moved to Moscow in 1912, where Urysohn completed his secondary education. While still at school, he worked at Shanyavsky Moscow City People's University, Shanyavsky University on an experimental p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doctor Of Sciences
A Doctor of Sciences, abbreviated д-р наук or д. н.; ; ; ; is a higher doctoral degree in the Russian Empire, Soviet Union and many Commonwealth of Independent States countries. One of the prerequisites of receiving a Doctor of Sciences degree is having a PhD-equivalent Candidate of Sciences degree beforehand. In addition, the Doctor of Sciences conferral also requires applicants to demonstrate significant and outstanding contributions to their research field. This degree is generally regarded as an honor and recognition for lifetime academic achievements rather than an ordinary academic degree by courses and theses. History The "Doctor of Sciences" degree was introduced in the Russian Empire in 1819 and abolished in 1917. Later it was revived in the USSR on January 13, 1934, by a decision of the Council of People's Commissars of the USSR. By the same decision, a lower degree, " Candidate of Sciences" (''kandidat nauk''), roughly the Russian equivalent to the research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steklov Institute Of Mathematics
Steklov Institute of Mathematics or Steklov Mathematical Institute () is a premier research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. The institute is named after Vladimir Andreevich Steklov, who in 1919 founded the Institute of Physics and Mathematics in Saint Petersburg, Leningrad. In 1934, this institute was split into separate parts for physics and mathematics, and the mathematical part became the Steklov Institute. At the same time, it was moved to Moscow. The first director of the Steklov Institute was Ivan Matveyevich Vinogradov. From 19611964, the institute's director was the notable mathematician Sergei Chernikov. The old building of the Institute in Leningrad became its Department in Leningrad. Today, that department has become a separate institute, called the ''St. Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences'' or PDMI RAS, located in Saint Petersburg, Russia. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Candidate Of Sciences
A Candidate of Sciences is a Doctor of Philosophy, PhD-equivalent academic research degree in all the post-Soviet countries with the exception of Ukraine, and until the 1990s it was also awarded in Central and Eastern European countries. It is officially classified by UNESCO as International Standard Classification of Education, ISCED level 8, "doctoral or equivalent". In those countries conferring the Candidate of Sciences degrees, a more advanced degree, Doctor of Sciences, is usually conferred as a higher doctorate. The Candidate of Sciences degree may be recognized as a Doctor of Philosophy#USSR, Russian Federation and former Soviet Republics, Doctor of Philosophy, usually in natural sciences, by scientific institutions in other countries. Overview The degree was introduced in the USSR on 13 January 1934 by a decision of the Government of the Soviet Union, Council of People's Commissars of the USSR, all previous degrees, ranks and titles having been abolished imme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, an open set is a member of a given Set (mathematics), collection of Subset, subsets of a given set, a collection that has the property of containing every union (set theory), union of its members, every finite intersection (set theory), intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology (structure), topology. These conditions are very loose, and allow enormous flexibility in the choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (topology)
In mathematics, a base (or basis; : bases) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open intervals in the real number line \R is a basis for the Euclidean topology on \R because every open interval is an open set, and also every open subset of \R can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X form a base for a topology on X. Under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
