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Bagpipe Theorem
In mathematics, the bagpipe theorem of describes the structure of the connected (but possibly non-paracompact) ω-bounded surfaces by showing that they are "bagpipes": the connected sum of a compact "bag" with several "long pipes". Statement A space is called ω-bounded if the closure of every countable set is compact. For example, the long line and the closed long ray are ω-bounded but not compact. When restricted to a metric space ω-boundedness is equivalent to compactness. The bagpipe theorem states that every ω-bounded connected surface is the connected sum of a compact connected surface and a finite number of long pipes. A space P is called a long pipe if there exist subspaces \ each of which is homeomorphic to S^1 \times \mathbb such that for n |
Chambers 1908 Bagpipe
Chambers may refer to: Places Canada: *Chambers Township, Ontario United States: *Chambers County, Alabama *Chambers, Arizona, an unincorporated community in Apache County *Chambers, Nebraska *Chambers, West Virginia *Chambers Township, Holt County, Nebraska *Chambers Branch, a stream in Kansas *Chambers County, Texas Other * ''Chambers Dictionary'' of the English Language * Chambers Harrap, the publishers of Chambers Dictionary * Chambers and Partners, a British organisation that produces international rankings for the legal industry * Chambers of parliament * ''Chambers'' (album), by Steady & Co. (2001) * Hedingham & Chambers, a bus company in Suffolk and Essex * judge's chambers, a judge's office where some matters are heard out of court * barristers' chambers, in some English-speaking countries a set of rooms from which barristers practice * ''Chambers'' (series), a BBC Radio 4 legal sitcom starring John Bird which later moved to television * Chambers stove, a defunct s ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paracompact Space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed 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 subspace be paracompact. Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ω-bounded Space
In mathematics, an ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if ''P'' is some property of subspaces, then a ''P''-bounded space is one in which every subspace with property ''P'' has compact closure. Every compact space is ω-bounded, and every ω-bounded space is countably compact. The long line is ω-bounded but not compact. The bagpipe theorem In mathematics, the bagpipe theorem of describes the structure of the connected (but possibly non-paracompact) ω-bounded surfaces by showing that they are "bagpipes": the connected sum of a compact "bag" with several "long pipes". Statement ... describes the ω-bounded surfaces. References * Properties of topological spaces {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces. More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots. Connected sum at a point A connected sum of two ''m''-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth catego ... [...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 by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''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 top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long Line (topology)
In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither Lindelöf nor separable). Therefore, it serves as one of the basic counterexamples of topology. Intuitively, the usual real-number line consists of a countable number of line segments [0,1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments. Definition The closed long ray L is defined as the cartesian product of the First uncountable ordinal, first uncountable ordinal \omega_1 with the Interval (mathematics), half-open interval [0, 1), equipped with the order topology that arises from the lexicographical order on \omega_1 \times [0,1). The open long ray is obtained from the closed long ray by removing the smallest element (0, 0). The long line is obtained by putting together a long ray i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prüfer Manifold
In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that is not paracompact. It was introduced by and named after Heinz Prüfer Ernst Paul Heinz Prüfer (10 November 1896 – 7 April 1934) was a German Jewish mathematician born in Wilhelmshaven. His major contributions were on abelian groups, graph theory, algebraic numbers, knot theory and Sturm–Liouville theory. In .... Construction The Prüfer manifold can be constructed as follows . Take an uncountable number of copies ''X''''a'' of the plane, one for each real number ''a'', and take a copy ''H'' of the upper half plane (of pairs (''x'', ''y'') with ''y'' > 0). Then glue the ''open upper half'' of each plane ''X''''a'' to the upper half plane ''H'' by identifying (''x'',''y'')∈''X''''a'' for ''y'' > 0 with the point in ''H''. The resulting quotient space Q is the Prüfer manifold. The images in Q of the points (0,0) of the spaces ''X' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surfaces
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. Surface or surfaces may also refer to: Mathematics *Surface (mathematics), a generalization of a plane which needs not be flat * Surface (differential geometry), a differentiable two-dimensional manifold *Surface (topology), a two-dimensional manifold * Algebraic surface, an algebraic variety of dimension two * Coordinate surfaces * Fractal surface, generated using a stochastic algorithm * Polyhedral surface *Surface area *Surface integral Arts and entertainment *Surface (band), an American R&B and pop trio ** ''Surface'' (Surface album), 1986 * Surfaces (band), American musical duo * ''Surface'' (Circle album), 1998 * "Surface" (Aero Chord song), 2014 * ''Surface'' (2005 TV series), an American science fiction show, 2005–2006 * ''Surface'' (2022 TV series), an American psychological thriller miniseries that began streaming in 2022 *''The Surface'', an American film ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
