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R H Bing
R. H. Bing (October 20, 1914 – April 28, 1986) was an American mathematician who worked mainly in the areas of geometric topology and continuum theory. His father was named Rupert Henry, but Bing's mother thought that "Rupert Henry" was too British for Texas. She compromised by abbreviating it to R. H. Consequently, R. H. does not stand for a first or middle name. Mathematical contributions Bing's mathematical research was almost exclusively in 3-manifold theory and in particular, the geometric topology of \mathbb R^3. The term Bing-type topology was coined to describe the style of methods used by Bing. Bing established his reputation early on in 1946, soon after completing his Ph.D. dissertation, by solving the Kline sphere characterization problem. In 1948 he proved that the pseudo-arc is homogeneous space, homogeneous, contradicting a published but erroneous 'proof' to the contrary. In 1951, he proved results regarding the metrizability of topological spaces, includi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oakwood, Texas
Oakwood is a town in Leon County, Texas, Leon and Freestone County, Texas, Freestone counties in the U.S. state of Texas. The population was 389 at the 2020 United States census, 2020 census. It was founded in 1872 as a stop on the International Railroad. Geography Oakwood is located in northeastern Leon County. U.S. Geological Survey, USGS topographic maps show the northern town boundary following the Leon/Freestone County line, but maps by the U.S. Census show the town extending slightly north into Freestone County. U.S. Route 79 in Texas, U.S. Route 79 passes through the town as Broad Street, leading northeast to Palestine, Texas, Palestine and southwest to Buffalo, Texas, Buffalo. According to the United States Census Bureau, the town has a total area of , all land. Demographics As of the 2020 United States census, there were 389 people, 236 households, and 193 families residing in the town. As of the census of 2000, there were 471 people, 199 households, and 131 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Space
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ''G'' are called the symmetries of ''X''. A special case of this is when the group ''G'' in question is the automorphism group of the space ''X'' – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, ''X'' is homogeneous if intuitively ''X'' looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of ''G'' be faithful (non-identity elements act non-trivially), althou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation of a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-ball
Three-ball (or "3-ball", colloquially) is a folk game of pool played with any three standard pool s and . The game is frequently gambled upon. The goal is to () the three object balls in as few shots as possible.PoolSharp's "Three-Ball Rules" The game involves a somewhat more significant amount of than either or , because of the disproportionate value of pocketing balls on the shot and increased difficulty of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by Henri Poincaré in 1904, the theorem concerns spaces that locally look like ordinary three-dimensional space but which are finite in extent. Poincaré hypothesized that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. Attempts to resolve the conjecture drove much progress in the field of geometric topology during the 20th century. The eventual proof built upon Richard S. Hamilton's program of using the Ricci flow to solve the problem. By developing a number of new techniques and results in the theory of Ricci flow, Grigori Perelman was able to modify and complete Hamilton's program. In papers posted to the arXiv reposi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Suspension Theorem
In geometric topology, the double suspension theorem of James W. Cannon () and Robert D. Edwards states that the double suspension ''S''2''X'' of a homology sphere ''X'' is a topological sphere.James W. Cannon, "Σ2 H3 = S5 / G", ''Rocky Mountain J. Math.'' (1978) 8, pp. 527-532. If ''X'' is a piecewise-linear homology sphere but not a sphere, then its double suspension ''S''2''X'' (with a triangulation derived by applying the double suspension operation to a triangulation of ''X'') is an example of a triangulation of a topological sphere that is not piecewise-linear. The reason is that, unlike in piecewise-linear manifolds, the link of one of the suspension points is not a sphere. See also * References * *{{Citation , last1=Latour , first1=François , title=Séminaire Bourbaki vol. 1977/78 Exposés 507–524 , publisher=Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Schoenflies Conjecture
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem. Original formulation The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane. An alternative statement is that if C \subset \mathbb R^2 is a simple closed curve, then there is a homeomorphism f : \mathbb R^2 \to \mathbb R^2 such that f(C) is the unit circle in the plane. Elementary proofs can be found in , , and . The result can first be proved for polygons when the homeomorphism can be taken to be piecewise linear and the identity map off some compact set; the case of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crumpled Cubes
In geometric topology, a branch of mathematics, a crumpled cube is any space in R3 homeomorphic to a 2-sphere A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ''center' ... together with its interior. Lininger showed in 1965 that the union of a crumpled cube and an open 3-ball glued along their boundaries is a 3-sphere. References Geometric topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smith Conjecture
In mathematics, the Smith conjecture states that if ''f'' is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of ''f'' cannot be a nontrivial knot. showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have a fixed point set equal to a circle, and asked in if the fixed point set could be knotted. proved the Smith conjecture for the special case of diffeomorphisms of order 2 (and hence any even order). The proof of the general case was described by and depended on several major advances in 3-manifold theory, In particular the work of William Thurston on hyperbolic structures on 3-manifolds, and results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, with some additional help from Bass, Cameron Gordon, Peter Shalen, and Rick Litherland. gave an example of a continuous involution of the 3-sphere whose fixed point set is a wildly embedded circle, so the Smith conjecture i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-sphere
A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ''center'' of the sphere, and the distance is the sphere's ''radius''. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions and is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: : a_0 = a_1 = 1 and a_n = a_ + (n - 1)a_ for n > 1. The first few terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-sphere
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior of a 3-sphere is a 4-ball. It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even though it is curved into the 4th dimension. For example, when traveling on a 3-sphere, you can go north and south, east and west, or along a 3rd set of cardinal directions. This means that a 3-sphere is an example of a 3-manifold. Definition In coordinates, a 3-sphere with center and radius is the set of all points in real, Four-dimensional space, 4-dimensional space () such that :\sum_^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 + ( x_1 - C_1 )^2 + ( x_2 - C_2 )^2+ ( x_3 - C_3 )^2 = r^2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted : :S^3 = \left\. It is often convenient to r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
