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N-connected Space
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of ''n''-connectedness generalizes the concepts of path-connectedness and simple connectedness. An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is ''n''-connected (or ''n''-simple connected) if its first ''n'' homotopy groups are trivial. Homotopical connectivity is defined for maps, too. A map is ''n''-connected if it is an isomorphism "up to dimension ''n,'' in homotopy". Definition using holes All definitions below consider a topological space ''X''. A hole in ''X'' is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point., Section 4.3 Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be the region of space bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Fiber
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)'' is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f:A \to B. It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groups\cdots \to \pi_(B) \to \pi_n(\text(f)) \to \pi_n(A) \to \pi_n(B) \to \cdotsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangleC(f)_\bullet 1\to A_\bullet \to B_\bullet \xrightarrowgives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber. Construction The homotopy fiber has a simple description for a continuous map f:A \to B. If we replace f by a fibr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term ''hyper''sphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension , which means that they cannot be easily visualized. The -sphere is the setting for -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the ''radius'') from a given '' center'' point. Its interior, consisting of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any Loop (topology), loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Set
In mathematics, a point (topology), point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a Neighborhood (mathematics), neighborhood of that does not contain any other points of . This is equivalent to saying that the Singleton (mathematics), singleton is an open set in the topological space (considered as a subspace topology, subspace of ). Another equivalent formulation is: an element of is an isolated point of if and only if it is not a limit point of . If the space is a metric space, for example a Euclidean space, then an element of is an isolated point of if there exists an open ball around that contains only finitely many elements of . A point set that is made up only of isolated points is called a discrete set or discrete point set (see also discrete space). Related notions Any discrete subset of Euclidean space must be countable, since the isolation of each of its points together with the fact that R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map (mathematics)
In mathematics, a map or mapping is a function (mathematics), function in its general sense. These terms may have originated as from the process of making a map, geographical map: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term ''transformation'' can be used interchangeably, but ''transformation (function), transformation'' often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. Maps as functions In many branches of mathematics, the term ''map'' is used to mean a Function (mathematics), function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Path
Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous game, a generalization of games used in game theory ** Law of continuity, a heuristic principle of Gottfried Leibniz * Continuous function, in particular: ** Continuity (topology), a generalization to functions between topological spaces ** Scott continuity, for functions between posets ** Continuity (set theory), for functions between ordinals ** Continuity (category theory), for functors ** Graph continuity, for payoff functions in game theory * Continuity theorem may refer to one of two results: ** Lévy's continuity theorem, on random variables ** Kolmogorov continuity theorem, on stochastic processes * In geometry: ** Parametric continuity, for parametrised curves ** Geometric continuity, a concept primarily applied to the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path-connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topological space X the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Group
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups were studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a construct that can be defined on a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional anal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointed Set
In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a Set (mathematics), set and x_0 is an element of X called the base point (also spelled basepoint). Map (mathematics), Maps between pointed sets (X, x_0) and (Y, y_0)—called based maps, pointed maps, or point-preserving maps—are function (mathematics), functions from X to Y that map one basepoint to another, i.e. maps f \colon X \to Y such that f(x_0) = y_0. Based maps are usually denoted f \colon (X, x_0) \to (Y, y_0). Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set X together with a single nullary operation *: X^0 \to X, which picks out the basepoint. Pointed maps are the homomorphisms of these algebraic structures. The Class (set theory), class of all pointed sets together with the class of all based maps forms a category theory, category. Every pointed set can be converted to an ordinary set by forgetting the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Group
In mathematics, a trivial group or zero group is a group that consists of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: , , or depending on the context. If the group operation is denoted then it is defined by . The similarly defined is also a group since its only element is its own inverse, and is hence the same as the trivial group. The trivial group is distinct from the empty set, which has no elements, hence lacks an identity element, and so cannot be a group. Definitions Given any group , the group that consists of only the identity element is a subgroup of , and, being the trivial group, is called the of . The term, when referred to " has no nontrivial proper subgroups" refers to the only subgroups of being the trivial group and the group itself. Properties The trivial group is cyclic of order ; as such it may be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
