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Ki / Vi . Ki / Vi for all i 2 I and . x/ . Ki /: The strong topology has all possible sets of this form for a base. X; Y /. We refer the reader to [Hirsch 1976], from which we have freely borrowed. 9. Let X D T N and X 0 D T 0 N 0 be trivial foliated spaces 0 (with T Rp , T 0 Rp ). X; X 0 /. Proof. Since all functions preserve leaves we may assume that X 0 D T 0 D ޒn , regarded as a foliated space with one leaf. X; ޒn / in the strong topology. Let fV˛ g be a locally fi nite open cover of X and for each ˛ let ˛ > 0.

Assume that the isotropy group at x; fg 2 G j gx D xg, has dimension independent of x. Then M is foliated by the orbits of G. (If 1 In the fi rst edition we relied upon [Lawson 1977] as a basic reference for foliations, and this is still a valuable source. The more recent books [Candel and Conlon 2000; 2003] give an extensive and thorough treatment of the geometric and topological structure of foliated manifolds and foliated spaces. The books and their authors have been very helpful to us in the preparation of the second edition.

More generally, if V is any fi nite-dimensional subspace of H , let us choose an orthonormal basis P '1 ; : : : ; 'n for V . x/ iD1 is independent of all choices. V / can be thought of as describing how the total dimension of V is distributed or localized over the space X . x/; iD1 where the inner product is taken pointwise in Hx , is a signed measure of total mass equal to the trace of T and which again describes how this total trace is distributed over the space X . V / is the orthogonal projection onto 16 I.