|
Local Finiteness
The term locally finite has a number of different meanings in mathematics: *Locally finite collection of sets in a topological space *Locally finite group *Locally finite measure * Locally finite operator in linear algebra *Locally finite poset *Locally finite space In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood, that is, an open neighborhood consisting of finitely many elements. A locally finite space is Alexandrov. A T1 s ..., a topological space in which each point has a finite neighborhood * Locally finite variety in the sense of universal algebra {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Finite Collection
In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension. A collection of subsets of a topological space X is said to be locally finite if each point in the space has a neighbourhood that intersects only finitely many of the sets in the collection. Note that the term locally finite has different meanings in other mathematical fields. Examples and properties A finite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite: for example, the collection of all subsets of \mathbb of the form (n, n+2) for an integer n. A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of \mathbb of the form (-n, n) for a natural number ''n''. If a collection of sets is locally finite, the collection of all closures of these sets is also locally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Finite Group
In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studied. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov. Definition and first consequences A locally finite group is a group for which every finitely generated subgroup is finite. Since the cyclic subgroups of a locally finite group are finitely generated hence finite, every element has finite order, and so the group is periodic. Examples and non-examples Examples: * Every finite group is locally finite * Every infinite direct sum of finite groups is locally finite (Although the direct product may not be.) * Omega-categorical groups * The Prüfer groups are locally finite abelian groups * Every Hamiltonian group is locally finite * Every periodic solvable group is locally finite . * Every s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Finite Measure
In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure. Definition Let (X, T) be a Hausdorff topological space and let \Sigma be a \sigma-algebra on X that contains the topology T (so that every open set is a measurable set, and \Sigma is at least as fine as the Borel \sigma-algebra on X). A measure/signed measure/complex measure \mu defined on \Sigma is called locally finite if, for every point p of the space X, there is an open neighbourhood N_p of p such that the \mu-measure of N_p is finite. In more condensed notation, \mu is locally finite if and only if \text p \in X, \text N_p \in T \mbox p \in N_p \mbox \left, \mu\left(N_p\right)\ < + \infty. Examples # Any on is locally finite, since it assigns u ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Finite Operator
In mathematics, a linear operator f: V\to V is called locally finite if the space V is the union of a family of finite-dimensional f-invariant subspaces. In other words, there exists a family \ of linear subspaces of V, such that we have the following: * \bigcup_ V_i=V * (\forall i\in I) f _isubseteq V_i * Each V_i is finite-dimensional. An equivalent condition only requires V to be the spanned by finite-dimensional f-invariant subspaces. If V is also a Hilbert space, sometimes an operator is called locally finite when the sum of the \ is only dense in V. Examples * Every linear operator on a finite-dimensional space is trivially locally finite. * Every diagonalizable (i.e. there exists a basis of V whose elements are all eigenvectors of f) linear operator is locally finite, because it is the union of subspaces spanned by finitely many eigenvectors of f. * The operator on \mathbb /math>, the space of polynomials with complex coefficients, defined by T(f(x))=xf(x), is ''not'' l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Finite Poset
In mathematics, a locally finite poset is a partially ordered set ''P'' such that for all ''x'', ''y'' ∈ ''P'', the interval 'x'', ''y''consists of finitely many elements. Given a locally finite poset ''P'' we can define its '' incidence algebra''. Elements of the incidence algebra are functions ''ƒ'' that assign to each interval 'x'', ''y''of ''P'' a real number ''ƒ''(''x'', ''y''). These functions form an associative algebra with a product defined by : (f * g)(x,y):=\sum_ f(x,z) g(z,y). There is also a definition of ''incidence coalgebra''. In theoretical physics a locally finite poset is also called a causal set and has been used as a model for spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe .... References Stanley, Richard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Finite Space
In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood, that is, an open neighborhood consisting of finitely many elements. A locally finite space is Alexandrov. A T1 space is locally finite if and only if it is discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a .... References * General topology Properties of topological spaces {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |