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Exhaustion By Compact Sets
In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space X is a nested sequence of compact subsets K_i of X (i.e. K_1\subseteq K_2\subseteq K_3\subseteq\cdots), such that each K_i is contained in the interior of K_, i.e. K_i\subset\text(K_), and X=\bigcup_^\infty K_i. A space admitting an exhaustion by compact sets is called exhaustible by compact sets. As an example, for the space X=\mathbb^n, the sequence of closed balls K_i = \ forms an exhaustion of the space by compact sets. There is a weaker condition that drops the requirement that K_i is in the interior of K_, meaning the space is σ-compact (i.e., a countable union of compact subsets.) Construction If there is an exhaustion by compact sets, the space is necessarily locally compact (if Hausdorff). The converse is also often true. For example, for a locally compact Hausdorff space X that is a countable union of compact subsets, we can construct an exhaustion as fo ... [...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] |
Hemicompact
In mathematics, in the field of topology, a Hausdorff topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in the sequence. This forces the union of the sequence to be the whole space, because every point is compact and hence must lie in one of the compact sets. Examples * Every compact space is hemicompact. * The real line is hemicompact. * Every locally compact Lindelöf space is hemicompact. Properties Every hemicompact space is σ-compactWillard 2004, p. 126 and if in addition it is first countable then it is locally compact. If a hemicompact space is weakly locally compact, then it is exhaustible by compact sets. Applications If X is a hemicompact space, then the space C(X, M) of all continuous functions f : X \to M to a metric space (M, \delta) with the compact-open topology is metrizable. To see this, take a sequence K_1,K_2,\dots of compact subsets of X such that ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactness (mathematics)
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 such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reinhold Remmert
Reinhold Remmert (22 June 1930 – 9 March 2016) was a German mathematician. Born in Osnabrück, Lower Saxony, he studied mathematics, mathematical logic and physics in Münster. He established and developed the theory of complex-analytic spaces in joint work with Hans Grauert. Until his retirement in 1995, he was a professor for complex analysis in Münster. Remmert wrote two books on number theory and complex analysis, which contain a huge amount of historical information together with references on important papers in the subject. See also * Remmert–Stein theorem Important publications * * References * Short biographyhosted at University of Münster The University of Münster (, until 2023 , WWU) is a public research university located in the city of Münster, North Rhine-Westphalia in Germany. With more than 43,000 students and over 120 fields of study in 15 departments, it is Germany's ... List of doctoral students 20th-century German mathematician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Grauert
Hans Grauert (8 February 1930 in Haren, Emsland, Germany – 4 September 2011) was a German mathematician. He is known for major works on several complex variables, complex manifolds and the application of sheaf theory in this area, which influenced later work in algebraic geometry.Bauer, I. C. ''et al.'' (2002Complex geometry: collection of papers dedicated to Hans Grauert Springer. Together with Reinhold Remmert he established and developed the theory of complex-analytic spaces. He became professor at the University of Göttingen in 1958, as successor to C. L. Siegel. The lineage of this chair traces back through an eminent line of mathematicians: Weyl, Hilbert, Riemann, and ultimately to Gauss.Grauert, H. (1994Selected Papers Springer. Until his death, he was professor emeritus at Göttingen. Grauert was awarded a fellowship of the Leopoldina and the von Staudt Prize. Early life Grauert attended school at the Gymnasium in Meppen before studying for a semester at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leon Ehrenpreis
Eliezer 'Leon' Ehrenpreis (May 22, 1930 – August 16, 2010, Brooklyn) was a mathematician at Temple University who proved the Malgrange–Ehrenpreis theorem, the fundamental theorem about differential operators with constant coefficients. He previously held tenured positions at Yeshiva University and at the Courant Institute at New York University. Early life and education Leon was born in New York City to a family of Jewish immigrants from Eastern Europe. He graduated from Stuyvesant High School and studied Mathematics as an undergraduate at City College of New York. Afterward, he enrolled as a doctoral student at Columbia University where he studied under mathematician Claude Chevalley, obtaining his PhD in 1953 at the age of 23. His doctoral thesis was entitled "Theory of Distributions in Locally Compact Spaces". Religion Ehrenpreis was also a Rabbi, having received his ordination from the renowned Rabbi Moshe Feinstein. He was the author of a work on the Chumash and other ... [...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 Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal space, normal, and a Hausdorff space is paracompact if and only if it admits partition of unity, partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed set, closed subspace (topology), 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 set, open subspace be paracompact. The notion of paracompact space is also studied in pointless topology, where it is more well-beh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Hausdorff Space
In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' have non-overlapping open neighborhoods. Thus ''p'' and ''C'' can be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means "a regular Hausdorff space". These conditions are examples of separation axioms. Definitions A topological space ''X'' is a regular space if, given any closed set ''F'' and any point ''x'' that does not belong to ''F'', there exists a neighbourhood ''U'' of ''x'' and a neighbourhood ''V'' of ''F'' that are disjoint. Concisely put, it must be possible to separate ''x'' and ''F'' with disjoint neighborhoods. A or is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T3 if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Appert Space
In general topology, a branch of mathematics, the Appert topology, named for , is a topology on the set of positive integers. In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almost every positive integer. The space ''X'' with the Appert topology is called the Appert space. Construction For a subset ''S'' of ''X'', let denote the number of elements of ''S'' which are less than or equal to ''n'': : \mathrm(n,S) = \#\ . ''S'' is defined to be open in the Appert topology if either it does not contain 1 or if it has asymptotic density equal to 1, i.e., it satisfies :\lim_ \frac = 1. The empty set is open because it does not contain 1, and the whole set ''X'' is open since \text(n,X)/n=1 for all ''n''. Related topologies The Appert topology is closely related to the Fort space topology that arises from giving the set of integers greater than one the discrete topology, and then taking the point 1 as the point at infin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
