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A metric space X is boundedly compact if all closed bounded subsets of X are compact. Every boundedly compact metric space is complete. (This is a generalization of the ...

A section of a fiber bundle gives an element of the fiber over every point in B. Usually it is described as a map s:B->E such that pi degreess is the identity on B. A ...

Let gamma(t) be a smooth curve in a manifold M from x to y with gamma(0)=x and gamma(1)=y. Then gamma^'(t) in T_(gamma(t)), where T_x is the tangent space of M at x. The ...

A map psi:M->M, where M is a manifold, is a finite-to-one factor of a map Psi:X->X if there exists a continuous surjective map pi:X->M such that psi degreespi=pi degreesPsi ...

An n-dimensional manifold M is said to be a homotopy sphere, if it is homotopy equivalent to the n-sphere S^n. Thus no homotopy group can distinguish between M and S^n. The ...

A homotopy from one embedding of a manifold M in N to another such that at every time, it is an embedding. The notion of isotopy is category independent, so notions of ...

The infinitesimal algebraic object associated with a Lie groupoid. A Lie algebroid over a manifold B is a vector bundle A over B with a Lie algebra structure [,] (Lie ...

Poincaré's lemma says that on a contractible manifold, all closed forms are exact. While d^2=0 implies that all exact forms are closed, it is not always true that all closed ...

Let x be a point in an n-dimensional compact manifold M, and attach at x a copy of R^n tangential to M. The resulting structure is called the tangent space of M at x and is ...

A theorem that classifies planar regular closed curves up to regular homotopy by their contour winding numbers (Whitney 1937). In his thesis, S. Smale generalized this result ...