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Homeomorphism Group
In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. They are important to the theory of topological spaces, generally exemplary of automorphism groups and topologically invariant in the group isomorphism sense. Properties and examples There is a natural group action of the homeomorphism group of a space on that space. Let X be a topological space and denote the homeomorphism group of X by G. The action is defined as follows: \begin G\times X &\longrightarrow X\\ (\varphi, x) &\longmapsto \varphi(x) \end This is a group action since for all \varphi,\psi\in G, \varphi\cdot(\psi\cdot x)=\varphi(\psi(x))=(\varphi\circ\psi)(x), where \cdot denotes the group action, and the identity element of G (which is the identity function on X) sends points to themselves. If this action is transitive, then the space is said to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact Space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mapping Class Group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Motivation Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of homeomorphisms from the space into itself, that is, continuous maps with continuous inverses: functions which stretch and deform the space continuously without breaking or gluing the space. This set of homeomorphisms can be thought of as a space itself. It forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The open sets of this new function space will be made up of sets of functions that map compact subsets ''K'' into open subsets ''U'' as ''K'' and ''U'' range throughout our original topological space, completed with their finite inters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Exact Sequence
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Image (mathematics), image of one morphism equals the kernel (algebra), kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i [...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 homeomorphic, but topological spaces that homeomorphic have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan. Introduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mapping Class Group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Motivation Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of homeomorphisms from the space into itself, that is, continuous maps with continuous inverses: functions which stretch and deform the space continuously without breaking or gluing the space. This set of homeomorphisms can be thought of as a space itself. It forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The open sets of this new function space will be made up of sets of functions that map compact subsets ''K'' into open subsets ''U'' as ''K'' and ''U'' range throughout our original topological space, completed with their finite inters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second parameter of ''H'' as time then ''H'' describes a ''continu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For example, the cyclic group of addition modulo ''n'' can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written , where G is the original group and N is the normal subgroup. This is read as '', where \text is short for modulo. (The notation should be interpreted w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Topology
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy, ''simple'' homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently. Differences between low-dimensional and high-dimensional topology Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. The editor-in-chief is Vadim Ponomarenko (San Diego State University). The journal gives the Lester R. Ford Award annually to "authors of articles of expository excellence" published in the journal. Editors-in-chief The following persons are or have been editor-in-chief: See also *''Mathematics Magazine'' *''Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Connected Space
In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting of open connected sets. As a stronger notion, the space ''X'' is locally path connected if every point admits a neighbourhood basis consisting of open path connected sets. Background Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of ''compact'' subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, ''connected'' subsets of \R^n (for ''n'' > 1) proved to be much more complicated. Indeed, while any compact Hausdorff sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
