dif8-page3 - iv) v (( u U ) n dom v ) = u v (U n dom v )....

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Unformatted text preview: iv) v (( u U ) n dom v ) = u v (U n dom v ). 44444 i i i e I i e I Definition. An n-dimensional topological manifold is a pair (X, A )------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ consisting of a point set X and a C-compatible atlas A on it, such that the topology induced by the atlas on X satisfies the following two conditions i) for any two distinct points x, y e X, one can find two disjoint neighborhoods of x and y (i.e. X is a Hausdorff space); ii) there exists a countable family of charts v , v , v ,... e A , the domain of 1 2 3 which cover X (X is a second countable topological space). Remark. In physics, the dimension of the configuration space of a------------------------------ mechanical system (provided that it is a manifold) is called the number of degrees of freedom....
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This note was uploaded on 01/08/2012 for the course MATH 203 taught by Professor Sedita during the Spring '09 term at SUNY Stony Brook.

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