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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, contradicting a published but erroneous 'proof' to the contrary. In 1951 he proved results regarding the metrizability of topological spaces, including what would late ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oakwood, Texas
Oakwood is a town in Leon and Freestone counties in the U.S. state of Texas. The population was 389 at the 2020 census. It was founded in 1872 as a stop on the International Railroad. Geography Oakwood is located in northeastern Leon County at (31.584816, –95.850666). 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 runs through the town as Broad Street, leading northeast to Palestine and southwest to Buffalo. According to the United States Census Bureau, the town has a total area of , all of it 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 families residing in the town. The population density was 429.9 people per square mile (165.3/km2). There were 256 housi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nagata–Smirnov Metrization Theorem
The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (that is, -locally finite) basis. A topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. A collection in a space X is countably locally finite (or -locally finite) if it is the union of a countable family of locally finite collections of subsets of X. Unlike Urysohn's metrization theorem In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ..., which provides only a sufficient condition for metrizability, this theorem provides both a necess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Property P Conjecture
In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot (mathematics), knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing (non-trivial) Dehn surgery on the knot is not simply-connected. The conjecture states that all knots, except the unknot, have Property P. Research on Property P was started by R. H. Bing, who popularized the name and conjecture. This conjecture can be thought of as a first step to resolving the Poincaré conjecture, since the Lickorish–Wallace theorem says any closed, orientable 3-manifold results from Dehn surgery on a link. If a knot K \subset \mathbb^ has Property P, then one cannot construct a counterexample to the Poincaré conjecture by surgery along K. A proof was announced in 2004, as the combined result of efforts of mathematicians working in several different fields. Algebraic Formulation Let [l], [m] \in \pi_(\mathbb^ \setminus K) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphic
In the mathematical field of topology, a homeomorphism, 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. The word ''homeomorphism'' comes from the Greek words '' ὅμοιος'' (''homoios'') = similar or same and '' μορφή'' (''morphē'') = shape or form, introduced to mathematics by Henri Poincaré in 1895. Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending 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 descr ... [...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 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 doing so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), 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 Grigori Perelman's theorem concerns spaces that locally look like ordinary Euclidean space, three-dimensional space but which are finite in extent. Poincaré hypothesized that if such a space has the additional property that each path (topology), loop in the space can be continuously tightened to a point, then it is necessarily a 3-sphere, three-dimensional sphere. Attempts to resolve the conjecture drove much progress in the field of geometric topology during the 20th century. The Perelman's proof built upon Richard S. Hamilton's ideas of using the Ricci flow to solve the problem. By developing a number of breakthrough new techniques and results in the theory of Ricci ... [...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 pe ... [...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] |
Bing Shrinking
In geometric topology, a branch of mathematics, the Bing shrinking criterion, introduced by , is a method for showing that a quotient of a space is homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ... to the space. References * Geometric topology {{topology-stub ... [...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 together with its interior. Lininger showed in 1965 that the union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ... of a crumpled cube and an open 3-ball glued along their boundaries is a 3-sphere. *{{topology-stub References Geometric topology ... [...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 Order (group theory), finite order, then the fixed point set of ''f'' cannot be a nontrivial knot (mathematics), 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 3-manifold, hyperbolic structures on 3-manifolds, and results by William Hamilton Meeks, III, William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, with some additional help from Bass, Cameron Gordon (mathematician), Cameron Gordon, Peter Shalen, and Rick Litherland. gave an example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. 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 centre of the sphere, and 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 object 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 earlier is the sphere's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
