By Abraham R., Marsden J.E.

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The cases A, B, or C= (ZI are easily checked. As metric spaces are normal, it follows easily that on the closed subsets of S , d is a metric. For further details, see Hausdorff [1962],J. Kelley [1975,p. 1311, or Michael [1951,p. 1521. Continuity of a map f: ~ - can 2 ~be rephrased as follows. 36 Proposition. Let S be a metric space and 2'. Then f: ~ 4 is continuous 2 ~ at uoE S iff for all that d (u, uo)< 6 implies: d the Hausdorff E metric on >0 there is a 6 >0 such ( i ) for all a Ef ( u ) , there is a b Ef (uo) such that d (a, b ) < E; that is and (ii) for all b E f (u,), there is an a E f ( u ) such that d(b, a)<&; that is, f ('0) c aEf(U)DE(a)' This proposition follows at once from the definitions of continuity and the Hausdorff metric.

The unique topology in which evely point discrete topology (O = 2,, the collection of all subsets). 8 = (0,S ) is called the trivial topology. (cl(A))is dense in S. u E S is called isolated is isolated is called the The topology in which and is called nowhere Thus, A is nowhere dense iff int(c1( A ) )= 0. 6 Definition. A topological space S is called Hausdoiff iff each two distinct points have disjoint neighborhoods (that is, with empty intersection). Similarly, S is called normal i f f each two disjoint closed sets have disjoint neighborhoods.

1441. 9 reduces to the usual product rule for derivatives. It will also be convenient to consider partial derivatives in this context. 10 Definition. Let U, c El, U, c E2 be open, and supposef: U, X U2+F and f is differentiable. e1 l, = Df(ul, u2)- (el, 0) We similarly define D2f. 1 1 Definition. A map f: U c E-+ V c F(U, V open) is a C r diffeomorphism i f f f is of class Cr, is a bijection (that is, one-to-one and onto V ) , and f is also of class Cr. The following is a major result that will be used many times later in the book.

### Foundations of Mechanics by Abraham R., Marsden J.E.

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