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A topological space X such that for every closed subset C of X and every point x in X\C, there is a continuous function f:X->[0,1] such that f(x)=0 and f(C)={1}. This is the ...

A vector field is a section of its tangent bundle, meaning that to every point x in a manifold M, a vector X(x) in T_xM is associated, where T_x is the tangent space.

A special case of a flag manifold. A Grassmann manifold is a certain collection of vector subspaces of a vector space. In particular, g_(n,k) is the Grassmann manifold of ...

Calabi-Yau spaces are important in string theory, where one model posits the geometry of the universe to consist of a ten-dimensional space of the form M×V, where M is a four ...

A basis for the real numbers R, considered as a vector space over the rationals Q, i.e., a set of real numbers {U_alpha} such that every real number beta has a unique ...

The word basis can arise in several different contexts. Speaking in general terms, an object is "generated" by a basis in whatever manner is appropriate. For example, a ...

The number of regions into which space can be divided by n mutually intersecting spheres is N=1/3n(n^2-3n+8), giving 2, 4, 8, 16, 30, 52, 84, ... (OEIS A046127) for n=1, 2, ...

If two single-valued continuous functions kappa(s) (curvature) and tau(s) (torsion) are given for s>0, then there exists exactly one space curve, determined except for ...

The maximal number of regions into which space can be divided by n planes is f(n)=1/6(n^3+5n+6) (Yaglom and Yaglom 1987, pp. 102-106). For n=1, 2, ..., these give the values ...

A mathematical structure (e.g., a group, vector space, or smooth manifold) in a category.