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Continuum (topology)
In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua. Definitions * A continuum that contains more than one point is called nondegenerate. * A subset ''A'' of a continuum ''X'' such that ''A'' itself is a continuum is called a subcontinuum of ''X''. A space homeomorphic to a subcontinuum of the Euclidean plane R2 is called a planar continuum. * A continuum ''X'' is homogeneous if for every two points ''x'' and ''y'' in ''X'', there exists a homeomorphism ''h'': ''X'' → ''X'' such that ''h''(''x'') = ''y''. * A Peano continuum is a continuum that is locally connected at each point. * An indecomposable continuum is a continuum that cannot be represented as the union of two proper subcontinua. A continuum ''X'' is hereditarily indecomposable if every subcontinuum of ''X'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical
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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or ''holes'', of a topological space. To define the ''n''-th homotopy group, the base-point-preserving maps from an ''n''-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the ''n''-th homotopy group, \pi_n(X), of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never equivalent ( homeomorphic), but topological spaces that homeomorphic have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shape Theory (mathematics)
Shape theory is a branch of topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory. Background Shape theory was reinvented, further developed and promoted by the Polish mathematician Karol Borsuk in 1968. Actually, the name ''shape theory'' was coined by Borsuk. Warsaw Circle Borsuk lived and worked in Warsaw, hence the name of one of the fundamental examples of the area, the Warsaw circle. It is a compact subset of the plane produced by "closing up" a topologist's sine curve with an arc. The homotopy groups of the Warsaw circle are all trivial, just like those of a point, and so any map between the Warsaw circle and a point induces a weak homotopy equivalence. However these two spaces are not homotopy equivalent. So by the Whitehead theorem, the Warsa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Menger Sponge
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension. Construction The construction of a Menger sponge can be described as follows: # Begin with a cube. # Divide every face of the cube into nine squares, like Rubik's Cube. This sub-divides the cube into 27 smaller cubes. # Remove the smaller cube in the middle of each face, and remove the smaller cube in the center of the more giant cube, leaving 20 smaller cubes. This is a level-1 Menger sponge (resembling a void cube). # Repeat steps two and three for each of the remaining smaller cubes, and continue to iterate ''ad infinitum''. The second iteration gives a level-2 sponge, the third iteration gives a level-3 sponge, and s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Continuum
In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set ''S'' of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound. More symbolically: ''S'' has the least upper bound property, and For each ''x'' in ''S'' and each ''y'' in ''S'' with ''x'' < ''y'', there exists ''z'' in ''S'' such that ''x'' < ''z'' < ''y'' A has the least upper bound property, if every nonempty subset of the set that is bounded above has a least upper bound in the set. Linear continua are particularly important in the field of |
Inverse Limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit in category theory. By working in the dual category, that is by reverting the arrows, an inverse limit becomes a direct limit or ''inductive limit'', and a ''limit'' becomes a colimit. Formal definition Algebraic objects We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (I, \leq) be a directed poset (not all authors require ''I'' to be directed). Let (''A''''i'')''i''∈''I'' be a family of groups and suppose we have a family of homomorphisms f_: A_j \to A_i for all i \leq j (note the order) with the following properties: # f_ is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-arc
In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R. H. Bing proved that, in a certain well-defined sense, most continua in R''n'', ''n'' ≥ 2, are homeomorphic to the pseudo-arc. History In 1920, Bronisław Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane R2 must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in R2 that is homeomorphic to each of its nondegenerate subcontinua must be an arc. In 1922, Knaster discovered the first example of a hereditarily indecomposable continuum ''K'', later named the pseudo-arc, giving a negative answer to a Mazurkiewicz question. In 1948, R. H. Bing proved that Knaster's continuum is homogeneous, i.e. for any two of its points there is a homeomorphism takin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solenoid (mathematics)
:''This page discusses a class of topological groups. For the wrapped loop of wire, see Solenoid.'' In mathematics, a solenoid is a compact space, compact connected space, connected topological space (i.e. a continuum (topology), continuum) that may be obtained as the inverse limit of an inverse system of topological groups and continuous function, continuous homomorphisms :f_i: S_ \to S_i \quad \forall i \ge 0 where each S_i is a circle and ''f''''i'' is the map that uniformly wraps the circle S_ for n_ times (n_ \geq 2) around the circle S_i. This construction can be carried out geometrically in the three-dimensional Euclidean space R3. A solenoid is a one-dimensional homogeneous indecomposable continuum that has the structure of a compact topological group. Solenoids were first introduced by Leopold Vietoris, Vietoris for the n_i = 2 case, and by David van Dantzig, van Dantzig the n_i = n case, where n\geq 2 is fixed. Such a solenoid arises as a one-dimensional expanding att ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Cube
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below). Definition The Hilbert cube is best defined as the topological product of the intervals , 1/''n''for ''n'' = 1, 2, 3, 4, ... That is, it is a cuboid of countably infinite dimension, where the lengths of the edges in each orthogonal direction form the sequence \lbrace 1/n \rbrace_. The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval , 1 In other words, it is topologically indistinguishable from the unit cube of countably infinite dimension. Some authors use the term "Hilbert cube" to mean this Cartesian product instead of the product of the ,\frac/math>. If a point in the Hilbert cube ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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