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K-connected
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] |
Homological Connectivity
In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups. Definitions Background ''X'' is ''homologically-connected'' if its 0-th homology group equals Z, i.e. H_0(X)\cong \mathbb, or equivalently, its 0-th reduced homology group is trivial: \tilde(X)\cong 0. * For example, when ''X'' is a graph and its set of connected components is ''C'', H_0(X)\cong \mathbb^ and \tilde(X)\cong \mathbb^ (see graph homology). Therefore, homological connectivity is equivalent to the graph having a single connected component, which is equivalent to graph connectivity. It is similar to the notion of a connected space. ''X'' is ''homologically 1-connected'' if it is homologically-connected, and additionally, its 1-th homology group is trivial, i.e. H_1(X)\cong 0. * For example, when ''X'' is a connected graph with vertex-set ''V'' and edge-set ''E'', H_1(X) \cong \mathbb^. Therefore, homological 1-connectivity is equivale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hole With A 0-dimensional Boundary
A hole is an opening in or through a particular medium, usually a solid body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of engineering. Depending on the material and the placement, a hole may be an indentation in a surface (such as a hole in the ground), or may pass completely through that surface (such as a hole created by a hole puncher in a piece of paper). Types Holes can occur for a number of reasons, including natural processes and intentional actions by humans or animals. Holes in the ground that are made intentionally, such as holes made while searching for food, for replanting trees, or postholes made for securing an object, are usually made through the process of digging. Unintentional holes in an object are often a sign of damage. Potholes and sinkholes can damage human settlements. Holes can occur in a wide variety of materials, and at a wide range ... [...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 are used to 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 topolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Group
In mathematics, a trivial group or zero group is a group consisting 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: 0, 1, or e depending on the context. If the group operation is denoted \, \cdot \, then it is defined by e \cdot e = e. 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 G, the group consisting of only the identity element is a subgroup of G, and, being the trivial group, is called the of G. The term, when referred to "G has no nontrivial proper subgroups" refers to the only subgroups of G being the trivial group \ and the group G itself. Properties The trivial group is cyc ... [...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 and x_0 is an element of X called the base point, also spelled basepoint. Maps between pointed sets (X, x_0) and (Y, y_0) – called based maps, pointed maps, or point-preserving maps – are functions from X to Y that map one basepoint to another, i.e. a map f \colon X \to Y such that f(x_0) = y_0. This is 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 of all pointed sets together with the class of all based maps form a category. In this category the pointed singleton sets (\, a) are initial objects and terminal objects,Mac Lane (1998) p.26 i.e. they are zero objects. There is a faithful functor from p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Group
In mathematics, topological groups are logically 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 have been 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 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 analysis. Formal defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Path
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline is ��the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which ��will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image of an interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borsuk–Ulam Theorem
In mathematics, the Borsuk–Ulam theorem states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center. Formally: if f: S^n \to \R^n is continuous then there exists an x\in S^n such that: f(-x)=f(x). The case n=1 can be illustrated by saying that there always exist a pair of opposite points on the Earth's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space. The case n=2 is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures, assuming that both parameters vary continuously in space. The Borsuk–Ulam theorem has several equivalent statements in terms of odd functions. Recall that S^n is the ''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map (mathematics)
In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a 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'' 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, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a " continuous function" in topology, a " linear transformation" in linear algebra, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Set
] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equivalent to saying that the singleton is an open set in the topological space ''S'' (considered as a subspace of ''X''). Another equivalent formulation is: an element ''x'' of ''S'' is an isolated point of ''S'' if and only if it is not a limit point of ''S''. If the space ''X'' is a metric space, for example a Euclidean space, then an element ''x'' of ''S'' is an isolated point of ''S'' if there exists an open ball around ''x'' which contains only finitely many elements of ''S''. Related notions A set that is made up only of isolated points is called a discrete set (see also discrete space). Any discrete subset ''S'' of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
