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A subset F subset R of the real numbers is said to be an F_sigma set provided F is the countable union of closed sets. The name F_sigma comes from French: The F stands for ...

Let M^n be an n-manifold and let F={F_alpha} denote a partition of M^n into disjoint pathwise-connected subsets. Then F is called a foliation of M^n of codimension c (with ...

A subset G subset R of the real numbers is said to be a G_delta set provided G is the countable intersection of open sets. The name G_delta comes from German: The G stands ...

The generalized diameter is the greatest distance between any two points on the boundary of a closed figure. The diameter of a subset E of a Euclidean space R^n is therefore ...

There does not exist an everywhere nonzero tangent vector field on the 2-sphere S^2. This implies that somewhere on the surface of the Earth, there is a point with zero ...

The axioms formulated by Hausdorff (1919) for his concept of a topological space. These axioms describe the properties satisfied by subsets of elements x in a neighborhood ...

There are two possible definitions: 1. Possessing similarity of form, 2. Continuous, one-to-one, in surjection, and having a continuous inverse. The most common meaning is ...

A continuous transformation from one function to another. A homotopy between two functions f and g from a space X to a space Y is a continuous map G from X×[0,1]|->Y such ...

Let M be a Riemannian manifold, and let the topological metric on M be defined by letting the distance between two points be the infimum of the lengths of curves joining the ...

Let {f_n(x)} be a sequence of analytic functions regular in a region G, and let this sequence be uniformly convergent in every closed subset of G. If the analytic function ...