Search Results for ""

For any two points x,y in X there exists two open sets U and V such that x in U and y not in U, and y in V and x not in V. A space satisfying this axiom is known as a ...

Let M be a regular surface with v_(p),w_(p) points in the tangent space M_(p) of M. Then the third fundamental form is given by III(v_(p),w_(p))=S(v_(p))·S(w_(p)), where S is ...

The trivial loop is the loop that takes every point to its basepoint. Formally, if X is a topological space and x in X, the trivial loop based at x is the map L:[0,1]->X ...

A principal bundle is a special case of a fiber bundle where the fiber is a group G. More specifically, G is usually a Lie group. A principal bundle is a total space E along ...

A fundamental result of de Rham cohomology is that the kth de Rham cohomology vector space of a manifold M is canonically isomorphic to the Alexander-Spanier cohomology ...

An angle bracket is the combination of a bra and ket (bra+ket = bracket) which represents the inner product of two functions or vectors (or 1-forms), <f|g>=intf(x)g^|(x)dx in ...

An extreme point of a subset K of a vector space X is an extreme set S of K which consists of a single point x in K. The collection of all extreme points of K is sometimes ...

A decomposition of a module into a direct sum of submodules. The index set for the collection of submodules is then called the grading set. Graded modules arise naturally in ...

The term "homology group" usually means a singular homology group, which is an Abelian group which partially counts the number of holes in a topological space. In particular, ...

A generalization of an ordinary two-dimensional surface embedded in three-dimensional space to an (n-1)-dimensional surface embedded in n-dimensional space. A hypersurface is ...