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For X a topological space, the presheaf F of Abelian groups (rings, ...) on X is defined such that 1. For every open subset U subset= X, an Abelian group (ring, ...) F(U), ...

The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold C^* (C-star) which is the one-point compactification of the complex numbers ...

A species of structures is a rule F which 1. Produces, for each finite set U, a finite set F[U], 2. Produces, for each bijection sigma:U->V, a function F[sigma]:F[U]->F[V]. ...

A characterization of normal spaces which states that a topological space X is normal iff, for any two nonempty closed disjoint subsets A, and B of X, there is a continuous ...

A vector field is a map f:R^n|->R^n that assigns each x a vector f(x). Several vector fields are illustrated above. A vector field is uniquely specified by giving its ...

A fixed point is a point that does not change upon application of a map, system of differential equations, etc. In particular, a fixed point of a function f(x) is a point x_0 ...

A hedgehog is an envelope parameterized by its Gauss map (Martinez-Maure 1996). Viewed another way, a hedgehog is a Minkowski difference of a convex body (Martinez-Maure ...

A sequence which arises in the hypothetical reproduction of a population of rabbits. Let the substitution system map 0->1 correspond to young rabbits growing old, and 1->10 ...

A smooth structure on a topological manifold (also called a differentiable structure) is given by a smooth atlas of coordinate charts, i.e., the transition functions between ...

A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequence This commutative ...