# Introduction to modern topology and geometry, with homeworks by Katok A., Sossinsky A. By Katok A., Sossinsky A.

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If h : Y × [0, 1] → Y is a homotopy between the identity and a constant map of Y ,that is, h(y, 0) = y and h(y, 1) = y0 . Then for the map H := IdX ×h one has H(x, y, 0) = (x, y) and H(x, y, 1) = (x, y0 ). Thus the projection π1 : (x, y) → x and the embedding iy0 : x → (x, y0 ) provide a homotopy equivalence. 3. , those that are homotopy equivalent to a point. In this section, we consider the simplest type of space from the point of view of dimension and local structure: graphs, which may be described as onedimensional topological spaces consisting of line segments with some endpoints identified.

E. with all tangent vectors making non-zero angles with each other. In order to compute the index of p with respect to an immersed curve f , let us join p by a (nonclosed) smooth curve α transversal to f to a far away point a and move from a to p along that curve. , so that the tangent vector to f looks to the right of α) and subtract one when we cross it in the negative direction. When we reach the connected component of the complement to the curve containing p, we will obtain a certain integer i(p).

For example, instead of throwing out the middle one third intervals at each step, one can do it on the first step and then throw out in1 in the middle of two remaining interval and inductively tervals of length 18 throw out the interval of length 2n 31n+1 in the middle of each of 2n intervals which remain after n steps. 7. 1. 1. Prove (by computing the infinite sum of lengths of the deleted intervals) that the Cantor set C(1/3) has Lebesgue measure 0 ˆ although nowhere dense, (which was to be expected), whereas the set C, has positive Lebesgue measure.

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