By Y. Eliashberg
This ebook provides the 1st steps of a conception of confoliations designed to hyperlink geometry and topology of three-d touch constructions with the geometry and topology of codimension-one foliations on third-dimensional manifolds. constructing nearly independently, those theories in the beginning look belonged to 2 diverse worlds: the idea of foliations is a part of topology and dynamical structures, whereas touch geometry is the odd-dimensional 'brother' of symplectic geometry. although, either theories have constructed a couple of amazing similarities. Confoliations - which interpolate among touch buildings and codimension-one foliations - might help us to appreciate greater hyperlinks among the 2 theories. those hyperlinks supply instruments for transporting effects from one box to the other.It's gains comprise: a unified method of the topology of codimension-one foliations and make contact with geometry; perception at the geometric nature of integrability; and, new effects, particularly at the perturbation of confoliations into touch buildings
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On each axis the projections coordinate axis, for i = 1, 2, form a contracting sequence of intervals. Let xi be a number common to all the intervals formed by the projections on the i-th coordinate axis. Then the point x of Rm whose coordinates are ( X I , x2, * * * , xm) is a point of Bk for all k's. Since x E B , there is an open set U of the covering C such that x E U. V(x, r, B ) C U. Let d denote the length of the longest edge of B. Since all edges were bisected at each stage of the construction of the sequence, it follows that d/2k is the length of the longest edge of Bk.
2'. Both the empty set 0 and X itself are simuliamusly open sets of X and closed sets of X . An open set of X is usually not a closed set of X, and vice versa. In Section 7 we shall make a careful study of subsets of X which are both open and closed in X. 3'. The intersection of any collection of closed sets of X (finite or inJinite in number) is a closed set of X . 4'. If X C Rn, then the wllectwn of closed sets of X coincides with the collection of intersections of X with all closed sets of Rn.
An E m + ki 10 k2 - + - + . a * + - 1oa 0 ) . Since kn 10" ' + it is clear that an 5 c. Now the decimal expansionsof an 1/10, and of c agree out to the n-th digits, but the n-th digit of c is kn while COMPLETENESS OF REAL NUMBERS 851 that of an 35 + 1/10" is R, + 1. It follows that an 5 c 5 an+- 1 lo" - These inequalities assert that c E I,, and since they hold for every integer n, it follows that c lies in each interval of the sequence, hence in the intersection of all of them. This proves the theorem in case the sequence contracts regularly.